The representation of such a transformation in analytical form involves a system of equations,x′s= ƒs(x1, x2, ..., xn), (s = 1, 2, ..., n),expressing x′1, x′2, ..., x′n, the co-ordinates of the transformed point in terms of x1, x2, ..., xn, the co-ordinates of the original point. In these equations the functions ƒsare analytical functions of their arguments. Within a properly limited region they must be one-valued, and the equations must admit a unique solution with respect to x1, x2, ..., xn, since the operation would not otherwise be a definite one.From this point of view the operations of a continuous group, which depends on a set of r parameters, will be defined analytically by a system of equations of the formx′s= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),(i.)where a1, a2, ..., arrepresent the parameters. If this operation be represented by A, and that in which b1, b2, ..., brare the parameters by B, then the operation AB is represented by the elimination (assumed to be possible) of x′1, x′2, ..., x′nbetween the equations (i.) and the equationsx″s= ƒs(x′1, x′2, ..., x′n; b1, b2, ..., br), (s = 1, 2, ..., n).Since AB belongs to the group, the result of the elimination must bex″s= ƒs(x1, x2, ..., xn; c1, c2, ..., cr),where c1, c2, ..., crrepresent another definite set of values of the parameters. Moreover, since A−1belongs to the group, the result of solving equations (i.) with respect to x1, x2, ..., xnmust bexs= ƒs(x′1, x′2, ..., x′n; d1, d2, ..., dr), (s = 1, 2, ..., n).Conversely, if equations (i.) are such that these two conditions are satisfied, they do in fact define a finite continuous group.It will be assumed that the r parameters which enter in equations (i.) are independent,i.e.that it is impossible to choose r′ (< r) quantities in terms of which a1, a2, ..., arcanInfinitesimal operation of a continuous group.be expressed. Where this is the case the group will be spoken of as a “group of order r.” Lie uses the term “r-gliedrige Gruppe.” It is to be noticed that the word order is used in quite a different sense from that given to it in connexion with groups of finite order.In regard to equations (i.), which define the general operation of the group, it is to be noticed that, since the group contains the identical operation, these equations must for some definite set of values of the parameters reduce to x′1= x1, x′2= x2, ..., x′n= xn. This set of values may, without loss of generality, be assumed to be simultaneous zero values. For if i1, i2, ..., irbe the values of the parameters which give the identical operation, and if we writeas= is+ a, (s = 1, 2, ..., r),then zero values of the new parameters a1, a2, ..., argive the identical operation.To infinitesimal values of the parameters, thus chosen, will correspond operations which cause an infinitesimal change in each of the variables. These are called infinitesimal operations. The most general infinitesimal operation of the group is that given by the systemx′s− xs= δxs=∂ƒsδa1+∂ƒsδa2+ ... +∂ƒsδar, (s = 1, 2, ..., n),∂a1∂a2∂arwhere, in ∂ƒs/∂ai, zero values of the parameters are to be taken. Since a1, a2, ..., arare independent, the ratios of δa1, δa2, ..., δarare arbitrary. Hence the most general infinitesimal operation of the group may be written in the formδxs=(e1∂ƒs+ e2∂ƒs+ ... + er∂ƒs)δt, (s = 1, 2, ..., n),∂a1∂a2∂arwhere e1, e2, ..., erare arbitrary constants, and δt is an infinitesimal.If F(x1, x2, ..., xn) is any function of the variables, and if an infinitesimal operation of the group be carried out on the variables in F, the resulting increment of F will be∂Fδx1+∂Fδx2+ ... +∂Fδxn.∂x1∂x2∂xnIf the differential operator∂ƒ1∂+∂ƒ2∂+ ... +∂ƒn∂∂ai∂x1∂ai∂x2∂ai∂xnbe represented by Xi, (i = 1, 2, ..., r), then the increment of F is given by(e1X1+ e2X2+ ... + erXr) Fδt.When the equations (i.) defining the general operation of the group are given, the coefficients ∂ƒs/∂ai, which enter in these differential operators are functions of the variables which can be directly calculated.The differential operator e1X1+ e2X2+ ... + erXrmay then be regarded as defining the most general infinitesimal operation of the group. In fact, if it be for a moment represented by X, then (1 + δtX)F is the result of carrying out the infinitesimal operation on F; and by putting x1, x2, ..., xnin turn for F, the actual infinitesimal operation is reproduced. By a very convenient, though perhaps hardly justifiable, phraseology this differential operator is itself spoken of as the general infinitesimal operation of the group. The sense in which this phraseology is to be understood will be made clear by the foregoing explanations.We suppose now that the constants e1, e2, ..., erhave assigned values. Then the result of repeating the particular infinitesimal operation e1X1+ e2X2+ ... + erXror X an infinite number of times is some finite operation of the group. The effect of this finite operation on F may be directly calculated. In fact, if δt is the infinitesimal already introduced, thendF= X·F,d2F= X·X·F, ...dtdt2HenceF′ = F + tdF+t2+d2F+ ...dt1·2dt2= F + tX·F +t2X·X·F + ...1·2It must, of course, be understood that in this analytical representation of the effect of the finite operation on F it is implied that t is taken sufficiently small to ensure the convergence of the (in general) infinite series.When x1, x2, ... are written in turn for F, the system of equationsx′s= (1 + tX +t2X·X + ...)xs, (s = 1, 2, ..., n)1·2(ii.)represent the finite operation completely. If t is here regarded as a parameter, this set of operations must in themselves constitute a group, since they arise by the repetition of a single infinitesimal operation. That this is really the case results immediately from noticing that the result of eliminating F′ betweenF′ = F + tX·F +t2X·X·F + ...1·2andF″ = F′ + t′X·F′ +t′2X·X·F′ + ...1·2isF″ = F + (t + t′) X·F +(t + t′)2X·X·F + ...1·2The group thus generated by the repetition of an infinitesimal operation is called acyclicalgroup; so that a continuous group contains a cyclical subgroup corresponding to each of its infinitesimal operations.The system of equations (ii.) represents an operation of the group whatever the constants e1, e2, ..., ermay be. Hence if e1t, e2t, ..., ert be replaced by a1, a2, ..., arthe equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a form of the general equations for any operation of the group, and are equivalent to the equations (i.). The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x2∂/∂x, so that there is only a single variable. The relation between x′ and t is given by dx′/dt = x′2, with the condition that x′ = x when t = 0. This gives at once x′ = x/(1 − tx), which might also be obtained by the direct use of (ii.).When the finite equations (i.) of a continuous group of order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the group can be directly constructed, and that it contains rRelations between the infinitesimal operations of a finite continuous group.arbitrary constants. This is equivalent to saying that the group contains r linearly independent infinitesimal operations; and that the most general infinitesimal operation is obtained by combining these linearly with constant coefficients. Moreover, when any r independent infinitesimal operations of the group are known, it has been seen how the general finite operation of the group may be calculated. This obviously suggests that it must be possible to define the group by means of its infinitesimal operations alone; and it is clear that such a definition would lend itself more readily to some applications (for instance, to the theory of differential equations) than the definition by means of the finite equations.On the other hand, r arbitrarily given linear differential operators will not, in general, give rise to a finite continuous group of order r; and the question arises as to what conditions such a set of operators must satisfy in order that they may, in fact, be the independent infinitesimal operations of such a group.If X, Y are two linear differential operators, XY − YX is also a linear differential operator. It is called the “combinant” of X and Y (Lie uses the expressionKlammerausdruck) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as Jacobi’s)(X(YZ)) + (Y(ZX)) + (Z(XY)) = 0holds between them. Now it may be shown that any continuous group of which X, Y are infinitesimal operations contains also (XY) among its infinitesimal operations. Hence if r linearly independent operations X1, X2, ..., Xrgive rise to a finite continuous group of order r, the combinant of each pair must be expressible linearly in terms of the r operations themselves: that is, there must be a system of relations(XiXj) =Σk=rk=1cijkXk,where the c’s are constants. Moreover, from Jacobi’s identity and the identity (XY) + (YX) = 0 it follows that the c’s are subject to the relationsandcijt+ cjit= 0,Σs(cjkscist+ ckiscjst+ cijsckst) = 0(iii.)for all values of i, j, k and t.The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in orderDetermination of the distinct types of continuous groups of a given order.that X1, X2, ..., Xrmay generate, as infinitesimal operations, a continuous group of order r, are also sufficient.For the proof of this fundamental theorem see Lie’s works (cf. Lie-Engel, i. chap. 9; iii. chap. 25).If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, ..., Xrand Y1, Y2, ..., Yrcan be chosen, so that in the relations(XiXj) = ΣcijsXs, (YiYj) = Σ dijsYs,the constants cijsand dijsare the same for all values of i, j and s, the two groups are simply isomorphic, Xsand Ysbeing corresponding infinitesimal operations.Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very differentform; for instance, the number of variables for the one may be different from that for the other. They are, however, said to be of the sametype, in the sense that the laws according to which their operations combine are the same for both.The problem of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r3quantities cijswhich satisfy the relationscijt+ cijt= 0,Σscijscskt+ cjkscsit+ ckiscsjt= 0.for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, ..., Xrmay be replaced by any r independent linear functions of themselves, and the c’s will then be transformed by a linear substitution containing r2independent parameters. This, however, does not alter the type of group considered.For a single parameter there is, of course, only one type of group, which has been called cyclical.For a group of order two there is a single relation(X1X2) = αX1+ βX2.If α and β are not both zero, let α be finite. The relation may then be written (αX1+ βX2, α−1X2) = αX1+ βX2. Hence if αX1+ βX2= X′1, and α−1X2= X′2, then (X′1X′2) = X′1. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) = 0.Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744).A problem of fundamental importance in connexion with any givenSelf-conjugate subgroups. Integrable groups.continuous group is the determination of the self-conjugate subgroups which it contains. If X is an infinitesimal operation of a group, and Y any other, the general form of the infinitesimal operations which are conjugate to X isX + t(XY) +t2((XY)Y) + ....1·2Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (XY), ((XY)Y), ..., where for Y each infinitesimal operation of the group is taken in turn. Hence if X′1, X′2, ..., X′sare s linearly independent operations of the group which generate a self-conjugate subgroup of order s, then foreveryinfinitesimal operation Y of the group relations of the form(X′iY) = Σe=se=1aieX′e, (i = 1, 2, ..., s)must be satisfied. Conversely, if such a set of relations is satisfied, X′1, X′2, ..., X′sgenerate a subgroup of order s, which contains every operation conjugate to each of the infinitesimal generating operations, and is therefore a self-conjugate subgroup.A specially important self-conjugate subgroup is that generated by the combinants of the r infinitesimal generating operations. That these generate a self-conjugate subgroup follows from the relations (iii.). In fact,((XiXj) Xk) = Σscijs(XsXk).Of the ½r(r − 1) combinants not more than r can be linearly independent. When exactly r of them are linearly independent, the self-conjugate group generated by them coincides with the original group. If the number that are linearly independent is less than r, the self-conjugate subgroup generated by them is actually a subgroup;i.e.its order is less than that of the original group. This subgroup is known as the derived group, and Lie has called a groupperfectwhen it coincides with its derived group. A simple group, since it contains no self-conjugate subgroup distinct from itself, is necessarily a perfect group.If G is a given continuous group, G1the derived group of G, G2that of G1, and so on, the series of groups G, G1, G2, ... will terminate either with the identical operation or with a perfect group; for the order of Gs+1is less than that of Gsunless Gsis a perfect group. When the series terminates with the identical operation, G is said to be anintegrablegroup; in the contrary case G is callednon-integrable.If G is an integrable group of order r, the infinitesimal operations X1, X2, ..., Xrwhich generate the group may be chosen so that X1, X2, ..., Xr1, (r1< r) generate the first derived group, X1, X2, ..., Xr2, (r2< r1) the second derived group, and so on. When they are so chosen the constants cijsare clearly such that if rp< i ≤ rp+1, rq< j ≤ rq+1, p ≥ q, then cijsvanishes unless s ≤ rp+1.In particular the generating operations may be chosen so that cijsvanishes unless s is equal to or less than the smaller of the two numbers i, j; and conversely, if the c’s satisfy these relations, the group is integrable.A simple group, as already defined, is one which has no self-conjugate subgroup. It is a remarkable fact that the determinationSimple groups.of all distinct types of simple continuous groups has been made, for in the case of discontinuous groups and groups of finite order this is far from being the case. Lie has demonstrated the existence of four great classes of simple groups:—(i.) The groups simply isomorphic with the general projective group in space of n dimensions. Such a group is defined analytically as the totality of the transformations of the formx′s=as,1x1+ as,2x2+ ... + as,nxn+ as, n + 1, (s = 1, 2, ..., n),an+1,1x1+ an+1,2x2+ ... + an+1,nxn+ 1where the a’s are parameters. The order of this group is clearly n(n + 2).(ii.) The groups simply isomorphic with the totality of the projective transformations which transform a non-special linear complex in space of 2n − 1 dimensions with itself. The order of this group is n(2n + 1).(iii.) and (iv.) The groups simply isomorphic with the totality of the projective transformations which change a quadric of non-vanishing discriminant into itself. These fall into two distinct classes of types according as n is even or odd. In either case the order is ½n(n + 1). The case n = 3 forms an exception in which the corresponding group is not simple. It is also to be noticed that a cyclical group is a simple group, since it has no continuous self-conjugate subgroup distinct from itself.W. K. J. Killing and E. J. Cartan have separately proved that outside these four great classes there exist only five distinct types of simple groups, whose orders are 14, 52, 78, 133 and 248; thus completing the enumeration of all possible types.To prevent any misapprehension as to the bearing of these very general results, it is well to point out explicitly that there are no limitations on the parameters of a continuous group as it has been defined above. They are to be regarded as taking in general complex values. If in the finite equations of a continuous group the imaginary symbol does not explicitly occur, the finite equations will usually define a group (in the general sense of the original definition) when both parameters and variables are limited to real values. Such a group is, in a certain sense, a continuous group; and such groups have been considered shortly by Lie (cf. Lie-Engel, iii. 360-392), who calls themrealcontinuous groups. To these real continuous groups the above statement as to the totality of simple groups does not apply; and indeed, in all probability, the number of types ofrealsimple continuous groups admits of no such complete enumeration. The effect of limitation to real transformations may be illustrated by considering the groups of projective transformations which changex2+ y2+ z2− 1 = 0 and x2+ y2− z2− 1 = 0respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformationx′ = x, y′ = y, z′ = z√ (−1),the general continuous groups which transform the two quadrics respectively into themselves are simply isomorphic. This is not, however, the case for therealcontinuous groups. In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators. The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple.Among the groups isomorphic with a given continuous group thereThe adjunct group.is one of special importance which is known as theadjunctgroup. This is a homogeneous linear group in a number of variables equal to the order of the group, whose infinitesimal operations are defined by the relationsXi= Σi, scijsxi∂, (j = 1, 2, ..., r),∂xswhere cijsare the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves.That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants. It is thus found that (XpXq) = ΣscpqsXs. The X’s, however, are not necessarily linearly independent. In fact, the sufficient condition that ΣjajXjshould be identically zero is that Σjajcijsshould vanish for all values of i and s. Hence if the equations Σjajcijs= 0 for all values of i and s have r′ linearly independent solutions, only r − r′ of the X’s are linearly independent, and the isomorphism of the two groups is multiple. If Y1, Y2, ..., Yrare the infinitesimal operations of the given group, the equationsΣjajcijs= 0, (s, i = 1, 2, ..., r)express the condition that the operations of the cyclical group generated by ΣjajYishould be permutable with every operation of the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r′ each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations.We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made betweenContinuous groups of the line of the plane, and of three-dimensional space.two simply isomorphic groups. We proceed to discuss the classification and theory of groups when their form is regarded as essential; and this is a return to a more geometrical point of view.It is natural to begin with the projective groups, which are the simplest in form and at the same time are of supreme importance in geometry. The general projective group of the straight line is the group of order three given byx′ =ax + b,cx + d′where the parameters are the ratios of a, b, c, d. Sincex′3− x′2·x′ − x′1=x3− x2·x − x1x′3− x′1x′ − x′2x3− x1x − x2is an operation of the above form, the group is triply transitive. Every subgroup of order two leaves one point unchanged, and all such subgroups are conjugate. A cyclical subgroup leaves either two distinct points or two coincident points unchanged. A subgroup which either leaves two points unchanged or interchanges them is an example of a “mixed” group.The analysis of the general projective group must obviously increase very rapidly in complexity, as the dimensions of the space to which it applies increase. This analysis has been completely carried out for the projective group of the plane, with the result of showing that there are thirty distinct types of subgroup. Excluding the general group itself, every one of these leaves either a point, a line, or a conic section unaltered. For space of three dimensions Lie has also carried out a similar investigation, but the results are extremely complicated. One general result of great importance at which Lie arrives in this connexion is that every projective group in space of three dimensions, other than the general group, leaves either a point, a curve, a surface or a linear complex unaltered.Returning now to the case of a single variable, it can be shown that any finite continuous group in one variable is either cyclical or of order two or three, and that by a suitable transformation any such group may be changed into a projective group.The genesis of an infinite as distinguished from a finite continuous group may be well illustrated by considering it in the case of a single variable. The infinitesimal operations of the projective group in one variable are d/dx, x(d/dx), x2(d/dx). If these combined with x3(d/dx) betaken as infinitesimal operations from which to generate a continuous group among the infinitesimal operations of the group, there must occur the combinant of x2(d/dx) and x3(d/dx). This is x4(d/dx). The combinant of this and x2(d/dx) is 2x5(d/dx) and so on. Hence xr(d/dx), where r is any positive integer, is an infinitesimal operation of the group. The general infinitesimal operation of the group is therefore ƒ(x)(d/dx), where ƒ(x) is an arbitrary integral function of x.In the classification of the groups, projective or non-projective of two or more variables, the distinction between primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite family of curves ƒ(x, y) = C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not. In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F(x, y, z) = C, which are interchanged among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) = a, H(x, y, z) = b, which are so interchanged.In regard to primitive groups Lie has shown that any primitive group of the plane can, by a suitably chosen transformation, be transformed into one of three definite types of projective groups; and that any primitive group of space of three dimensions can be transformed into one of eight definite types, which, however, cannot all be represented as projective groups in three dimensions.The results which have been arrived at for imprimitive groups in two and three variables do not admit of any such simple statement.We shall now explain the conception of contact-transformations and groups of contact-transformations. This conception,Contact transformations.like that of continuous groups, owes its origin to Lie.From a purely analytical point of view a contact-transformation may be defined as a point-transformation in 2n + 1 variables, z, x1, x2, ..., xn, p1, p2, ..., pnwhich leaves unaltered the equation dz − p1dx1− p2dx2− ... − pndxn= 0. Such a definition as this, however, gives no direct clue to the geometrical properties of the transformation, nor does it explain the name given.In dealing with contact-transformations we shall restrict ourselves to space of two or of three dimensions; and it will be necessary to begin with some purely geometrical considerations. An infinitesimal surface-element in space of three dimensions is completely specified, apart from its size, by its position and orientation. If x, y, z are the co-ordinates of some one point of the element, and if p, q, −1 give the ratios of the direction-cosines of its normal, x, y, z, p, q are five quantities which completely specify the element. There are, therefore, ∞5surface elements in three-dimensional space. The surface-elements of a surface form a system of ∞2elements, for there are ∞2points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of ∞2elements, for there are ∞1points on the curve, and at each ∞1surface-elements containing the tangent to the curve at the point. Similarly the surface-elements which contain a given point clearly form a system of ∞2elements. Now each of these systems of ∞2surface-elements has the property that if (x, y, z, p, q) and (x + dx, y + dy, z + dz, p + dp, q + dq) are consecutive elements from any one of them, then dz − pdx − qdy = 0. In fact, for a system of the first kind dx, dy, dz are proportional to the direction-cosines of a tangent line at a point of the surface, and p, q, −1 are proportional to the direction-cosines of the normal. For a system of the second kind dx, dy, dz are proportional to the direction-cosines of a tangent to the curve, and p, q, −1 give the direction-cosines of the normal to a plane touching the curve; and for a system of the third kind dx, dy, dz are zero. Now the most general way in which a system of ∞2surface-elements can be given is by three independent equations between x, y, z, p and q. If these equations do not contain p, q, they determine one or more (a finite number in any case) points in space, and the system of surface-elements consists of the elements containing these points;i.e.it consists of one or more systems of the third kind.If the equations are such that two distinct equations independent of p and q can be derived from them, the points of the system of surface-elements lie on a curve. For such a system the equation dz − pdx − qdy = 0 will hold for each two consecutive elements only when the plane of each element touches the curve at its own point.If the equations are such that only one equation independent of p and q can be derived from them, the points of the system of surface-elements lie on a surface. Again, for such a system the equation dz − pdx − qdy = 0 will hold for each two consecutive elements only when each element touches the surface at its own point. Hence, when all possible systems of ∞2surface-elements in space are considered, the equation dz − pdx − qdy = 0 is characteristic of the three special types in which the elements belong, in the sense explained above, to a point or a curve or a surface.Let us consider now the geometrical bearing of any transformation x′ = ƒ1(x, y, z, p, q), ..., q′ = ƒ5(x, y, z, p, q), of the five variables. It will interchange the surface-elements of space among themselves, and will change any system of ∞2elements into another system of ∞2elements. A special system,i.e.a system which belongs to a point, curve or surface, will not, however, in general be changed into another special system. The necessary and sufficient condition that a special system should always be changed into a special system is that the equation dz′ − p′dx′ − q′dy′ = 0 should be a consequence of the equation dz − pdx − qdy = 0; or, in other words, that this latter equation should be invariant for the transformation.When this condition is satisfied the transformation is such as to change the surface-elements of a surface in general into surface-elements of a surface, though in particular cases they may become the surface-elements of a curve or point; and similar statements may be made with respect to a curve or point. The transformation is therefore a veritable geometrical transformation in space of three dimensions. Moreover, two special systems of surface-elements which have an element in common are transformed into two new special systems with an element in common. Hence two curves or surfaces which touch each other are transformed into two new curves or surfaces which touch each other. It is this property which leads to the transformations in question being called contact-transformations. It will be noticed that an ordinary point-transformation is always a contact-transformation, but that a contact-transformation (in space of n dimensions) is not in general a point-transformation (in space of n dimensions), though it may always be regarded as a point-transformation in space of 2n + 1 dimensions. In the analogous theory for space of two dimensions a line-element, defined by (x, y, p), where 1 : p gives the direction-cosines of the line, takes the place of the surface-element; and a transformation of x, y and p which leaves the equation dy − pdx = 0 unchanged transforms the ∞1line-elements, which belong to a curve, into ∞1line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch.One of the simplest instances of a contact-transformation that can be given is the transformation by reciprocal polars. By this transformation a point P and a plane p passing through it are changed into a plane p′ and a point P′ upon it;i.e.the surface-element defined by P, p is changed into a definite surface-element defined by P′, p′. The totality of surface-elements which belong to a (non-developable) surface is known from geometrical considerations to be changed into the totality which belongs to another (non-developable) surface. On the other hand, the totality of the surface-elements which belong to a curve is changed into another set which belong to a developable. The analytical formulae for this transformation, when the reciprocation is effected with respect to the paraboloid x2+ y2− 2z = 0, are x′ = p, y′ = q, z′ = px + qy − z, p′ = x, q′ = y. That this is, in fact, a contact-transformation is verified directly by noticing thatdz′ − p′dx′ − q′dy′ = −d (z − px − qy) − xdp − ydq = −(dz − pdx − qdy).A second simple example is that in which every surface-element is displaced, without change of orientation, normal to itself through a constant distance t. The analytical equations in this case are easily found in the formx′ = x +pt, y′ = y +qt, z′ = z −t,√(1 + p2+ q2)√(1 + p2+ q2)√(1 + p2+ q2)p′ = q, q′ = q.That this is a contact-transformation is seen geometrically by noticing that it changes a surface into a parallel surface. Every point is changed by it into a sphere of radius t, and when t is regarded as a parameter the equations define a cyclical group of contact-transformations.The formal theory of continuous groups of contact-transformations is, of course, in no way distinct from the formal theory of continuous groups in general. On what may be called the geometrical side, the theory of groups of contact-transformations has been developed with very considerable detail in the second volume of Lie-Engel.To the manifold applications of the theory of continuous groups in various branches of pure and applied mathematicsApplications of the theory of continuous groups.it is impossible here to refer in any detail. It must suffice to indicate a few of them very briefly. In some of the older theories a new point of view is obtained which presents the results in a fresh light, and suggests the natural generalization. As an example, the theory of the invariants of a binary form may be considered.If in the form ƒ = a0xn+ na1xn−1y + ... + anyn, the variables be subjected to a homogeneous substitutionx′ = αx + βy, y′ = γx + δy,(i.)and if the coefficients in the new form be represented by accenting the old coefficients, thena′0= a0αn+ a1nαn−1γ + ... + anγn,a′1= a0αn−1β + a1{(n−1) αn−2βγ + αn−1δ} + ... + anγn−1δ,· · · · ·a′n= a0βn+ a1nβn−1δ + ... + anδn;(ii.)and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which αδ − βγ = 1, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations y(∂/∂x) and x(∂/∂y). Hence withthe same limitations on α, β, γ, δ the totality of the substitutions (ii.) forms a simply isomorphic continuous group of order 3, which is generated by the two infinitesimal transformationsa0∂+ 2a1∂+ 3a1∂+ ... + nan − 1∂,∂a1∂a2∂a3∂anandna1∂+ (n − 1)a2∂+ (n − 2)a3∂+ ... + au∂.∂a0∂a1∂a2∂au−1The invariants of the binary form,i.e.those functions of the coefficients which are unaltered by all homogeneous substitutions on x, y of determinant unity, are therefore identical with the functions of the coefficients which are invariant for the continuous group generated by the two infinitesimal operations last written. In other words, they are given by the common solutions of the differential equationsa0∂ƒ+ 2a1∂ƒ+ 3a2∂ƒ+ ... = 0,∂a1∂a1∂a2na1∂ƒ+ (n − 1)a2∂ƒ+ (n − 2)a3∂ƒ+ ... = 0.∂a0∂a1∂a2Both this result and the method by which it is arrived at are well known, but the point of view by which we pass from the transformation group of the variables to the isomorphic transformation group of the coefficients, and regard the invariants as invariants rather of the group than of the forms, is a new and a fruitful one.The general theory of curvature of curves and surfaces may in a similar way be regarded as a theory of their invariants for the group of motions. That something more than a mere change of phraseology is here implied will be evident in dealing with minimum curves,i.e.with curves such that at every point of them dx2+ dy2+ dz2= 0. For such curves the ordinary theory of curvature has no meaning, but they nevertheless have invariant properties in regard to the group of motions.The curvature and torsion of a curve, which are invariant for all transformations by the group of motions, are special instances of what are known asdifferential invariants. If ξ(∂/∂x) + η(∂/∂y) is the general infinitesimal transformation of a group of point-transformations in the plane, and if y1, y2, ... represent the successive differential coefficients of y, the infinitesimal transformation may be written in the extended formξ∂+ η∂+ η1∂+ η2∂+ ...∂x∂y∂y1∂y2where η1δt, η2δt, ... are the increments of y1, y2, .... By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. Conversely, if an ordinary, differential equation ƒ(x, y, y1, y2, ...) = 0 admits the transformations of a continuous group,i.e.if the equation is unaltered when x and y undergo any transformation of the group, then ƒ(x, y, y1, y2, ...) or some multiple of it must be a differential invariant of the group. Hence it must be possible to find two independent differential invariants α, β of the group, such that when these are taken as variables the differential equation takes the form F(α, β, dβ/dα, d2β/dα2, ...) = 0. This equation in α, β will be of lower order than the original equation, and in general simpler to deal with. Supposing it solved in the form β = φ(α), where for α, β their values in terms of x, y, y1, y2, ... are written, this new equation, containing arbitrary constants, is necessarily again of lower order than the original equation. The integration of the original equation is thus divided into two steps. This will show how, in the case of an ordinary differential equation, the fact that the equation admits a continuous group of transformations may be taken advantage of for its integration.The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. It is quite impossible here to give any account of all that Lie and his followers have done in this direction. An entirely new mode of regarding the problem of the integration of a differential equation has been opened up, and in the classification that arises from it all those apparently isolated types of equations which in the older sense are said to be integrable take their proper place. It may, for instance, be mentioned that the question as to whether Monge’s method will apply to the integration of a partial differential equation of the second order is shown to depend on whether or not a contact-transformation can be found which will reduce the equation to either ∂2z/∂x2= 0 or ∂2z/∂x∂y = 0. It is in this direction that further advance in the theory of partial differential equations must be looked for. Lastly, it may be remarked that one of the most thorough discussions of the axioms of geometry hitherto undertaken is founded entirely upon the theory of continuous groups.
The representation of such a transformation in analytical form involves a system of equations,
x′s= ƒs(x1, x2, ..., xn), (s = 1, 2, ..., n),
expressing x′1, x′2, ..., x′n, the co-ordinates of the transformed point in terms of x1, x2, ..., xn, the co-ordinates of the original point. In these equations the functions ƒsare analytical functions of their arguments. Within a properly limited region they must be one-valued, and the equations must admit a unique solution with respect to x1, x2, ..., xn, since the operation would not otherwise be a definite one.
From this point of view the operations of a continuous group, which depends on a set of r parameters, will be defined analytically by a system of equations of the form
x′s= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
(i.)
where a1, a2, ..., arrepresent the parameters. If this operation be represented by A, and that in which b1, b2, ..., brare the parameters by B, then the operation AB is represented by the elimination (assumed to be possible) of x′1, x′2, ..., x′nbetween the equations (i.) and the equations
x″s= ƒs(x′1, x′2, ..., x′n; b1, b2, ..., br), (s = 1, 2, ..., n).
Since AB belongs to the group, the result of the elimination must be
x″s= ƒs(x1, x2, ..., xn; c1, c2, ..., cr),
where c1, c2, ..., crrepresent another definite set of values of the parameters. Moreover, since A−1belongs to the group, the result of solving equations (i.) with respect to x1, x2, ..., xnmust be
xs= ƒs(x′1, x′2, ..., x′n; d1, d2, ..., dr), (s = 1, 2, ..., n).
Conversely, if equations (i.) are such that these two conditions are satisfied, they do in fact define a finite continuous group.
It will be assumed that the r parameters which enter in equations (i.) are independent,i.e.that it is impossible to choose r′ (< r) quantities in terms of which a1, a2, ..., arcanInfinitesimal operation of a continuous group.be expressed. Where this is the case the group will be spoken of as a “group of order r.” Lie uses the term “r-gliedrige Gruppe.” It is to be noticed that the word order is used in quite a different sense from that given to it in connexion with groups of finite order.
In regard to equations (i.), which define the general operation of the group, it is to be noticed that, since the group contains the identical operation, these equations must for some definite set of values of the parameters reduce to x′1= x1, x′2= x2, ..., x′n= xn. This set of values may, without loss of generality, be assumed to be simultaneous zero values. For if i1, i2, ..., irbe the values of the parameters which give the identical operation, and if we write
as= is+ a, (s = 1, 2, ..., r),
then zero values of the new parameters a1, a2, ..., argive the identical operation.
To infinitesimal values of the parameters, thus chosen, will correspond operations which cause an infinitesimal change in each of the variables. These are called infinitesimal operations. The most general infinitesimal operation of the group is that given by the system
where, in ∂ƒs/∂ai, zero values of the parameters are to be taken. Since a1, a2, ..., arare independent, the ratios of δa1, δa2, ..., δarare arbitrary. Hence the most general infinitesimal operation of the group may be written in the form
where e1, e2, ..., erare arbitrary constants, and δt is an infinitesimal.
If F(x1, x2, ..., xn) is any function of the variables, and if an infinitesimal operation of the group be carried out on the variables in F, the resulting increment of F will be
If the differential operator
be represented by Xi, (i = 1, 2, ..., r), then the increment of F is given by
(e1X1+ e2X2+ ... + erXr) Fδt.
When the equations (i.) defining the general operation of the group are given, the coefficients ∂ƒs/∂ai, which enter in these differential operators are functions of the variables which can be directly calculated.
The differential operator e1X1+ e2X2+ ... + erXrmay then be regarded as defining the most general infinitesimal operation of the group. In fact, if it be for a moment represented by X, then (1 + δtX)F is the result of carrying out the infinitesimal operation on F; and by putting x1, x2, ..., xnin turn for F, the actual infinitesimal operation is reproduced. By a very convenient, though perhaps hardly justifiable, phraseology this differential operator is itself spoken of as the general infinitesimal operation of the group. The sense in which this phraseology is to be understood will be made clear by the foregoing explanations.
We suppose now that the constants e1, e2, ..., erhave assigned values. Then the result of repeating the particular infinitesimal operation e1X1+ e2X2+ ... + erXror X an infinite number of times is some finite operation of the group. The effect of this finite operation on F may be directly calculated. In fact, if δt is the infinitesimal already introduced, then
Hence
It must, of course, be understood that in this analytical representation of the effect of the finite operation on F it is implied that t is taken sufficiently small to ensure the convergence of the (in general) infinite series.
When x1, x2, ... are written in turn for F, the system of equations
(ii.)
represent the finite operation completely. If t is here regarded as a parameter, this set of operations must in themselves constitute a group, since they arise by the repetition of a single infinitesimal operation. That this is really the case results immediately from noticing that the result of eliminating F′ between
and
is
The group thus generated by the repetition of an infinitesimal operation is called acyclicalgroup; so that a continuous group contains a cyclical subgroup corresponding to each of its infinitesimal operations.
The system of equations (ii.) represents an operation of the group whatever the constants e1, e2, ..., ermay be. Hence if e1t, e2t, ..., ert be replaced by a1, a2, ..., arthe equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a form of the general equations for any operation of the group, and are equivalent to the equations (i.). The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x2∂/∂x, so that there is only a single variable. The relation between x′ and t is given by dx′/dt = x′2, with the condition that x′ = x when t = 0. This gives at once x′ = x/(1 − tx), which might also be obtained by the direct use of (ii.).
When the finite equations (i.) of a continuous group of order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the group can be directly constructed, and that it contains rRelations between the infinitesimal operations of a finite continuous group.arbitrary constants. This is equivalent to saying that the group contains r linearly independent infinitesimal operations; and that the most general infinitesimal operation is obtained by combining these linearly with constant coefficients. Moreover, when any r independent infinitesimal operations of the group are known, it has been seen how the general finite operation of the group may be calculated. This obviously suggests that it must be possible to define the group by means of its infinitesimal operations alone; and it is clear that such a definition would lend itself more readily to some applications (for instance, to the theory of differential equations) than the definition by means of the finite equations.
On the other hand, r arbitrarily given linear differential operators will not, in general, give rise to a finite continuous group of order r; and the question arises as to what conditions such a set of operators must satisfy in order that they may, in fact, be the independent infinitesimal operations of such a group.
If X, Y are two linear differential operators, XY − YX is also a linear differential operator. It is called the “combinant” of X and Y (Lie uses the expressionKlammerausdruck) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as Jacobi’s)
(X(YZ)) + (Y(ZX)) + (Z(XY)) = 0
holds between them. Now it may be shown that any continuous group of which X, Y are infinitesimal operations contains also (XY) among its infinitesimal operations. Hence if r linearly independent operations X1, X2, ..., Xrgive rise to a finite continuous group of order r, the combinant of each pair must be expressible linearly in terms of the r operations themselves: that is, there must be a system of relations
(XiXj) =Σk=rk=1cijkXk,
where the c’s are constants. Moreover, from Jacobi’s identity and the identity (XY) + (YX) = 0 it follows that the c’s are subject to the relations
and
cijt+ cjit= 0,Σs(cjkscist+ ckiscjst+ cijsckst) = 0
(iii.)
for all values of i, j, k and t.
The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in orderDetermination of the distinct types of continuous groups of a given order.that X1, X2, ..., Xrmay generate, as infinitesimal operations, a continuous group of order r, are also sufficient.
For the proof of this fundamental theorem see Lie’s works (cf. Lie-Engel, i. chap. 9; iii. chap. 25).
If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, ..., Xrand Y1, Y2, ..., Yrcan be chosen, so that in the relations
(XiXj) = ΣcijsXs, (YiYj) = Σ dijsYs,
the constants cijsand dijsare the same for all values of i, j and s, the two groups are simply isomorphic, Xsand Ysbeing corresponding infinitesimal operations.
Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very differentform; for instance, the number of variables for the one may be different from that for the other. They are, however, said to be of the sametype, in the sense that the laws according to which their operations combine are the same for both.
The problem of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r3quantities cijswhich satisfy the relations
cijt+ cijt= 0,Σscijscskt+ cjkscsit+ ckiscsjt= 0.
for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, ..., Xrmay be replaced by any r independent linear functions of themselves, and the c’s will then be transformed by a linear substitution containing r2independent parameters. This, however, does not alter the type of group considered.
For a single parameter there is, of course, only one type of group, which has been called cyclical.
For a group of order two there is a single relation
(X1X2) = αX1+ βX2.
If α and β are not both zero, let α be finite. The relation may then be written (αX1+ βX2, α−1X2) = αX1+ βX2. Hence if αX1+ βX2= X′1, and α−1X2= X′2, then (X′1X′2) = X′1. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) = 0.
Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744).
A problem of fundamental importance in connexion with any givenSelf-conjugate subgroups. Integrable groups.continuous group is the determination of the self-conjugate subgroups which it contains. If X is an infinitesimal operation of a group, and Y any other, the general form of the infinitesimal operations which are conjugate to X is
Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (XY), ((XY)Y), ..., where for Y each infinitesimal operation of the group is taken in turn. Hence if X′1, X′2, ..., X′sare s linearly independent operations of the group which generate a self-conjugate subgroup of order s, then foreveryinfinitesimal operation Y of the group relations of the form
(X′iY) = Σe=se=1aieX′e, (i = 1, 2, ..., s)
must be satisfied. Conversely, if such a set of relations is satisfied, X′1, X′2, ..., X′sgenerate a subgroup of order s, which contains every operation conjugate to each of the infinitesimal generating operations, and is therefore a self-conjugate subgroup.
A specially important self-conjugate subgroup is that generated by the combinants of the r infinitesimal generating operations. That these generate a self-conjugate subgroup follows from the relations (iii.). In fact,
((XiXj) Xk) = Σscijs(XsXk).
Of the ½r(r − 1) combinants not more than r can be linearly independent. When exactly r of them are linearly independent, the self-conjugate group generated by them coincides with the original group. If the number that are linearly independent is less than r, the self-conjugate subgroup generated by them is actually a subgroup;i.e.its order is less than that of the original group. This subgroup is known as the derived group, and Lie has called a groupperfectwhen it coincides with its derived group. A simple group, since it contains no self-conjugate subgroup distinct from itself, is necessarily a perfect group.
If G is a given continuous group, G1the derived group of G, G2that of G1, and so on, the series of groups G, G1, G2, ... will terminate either with the identical operation or with a perfect group; for the order of Gs+1is less than that of Gsunless Gsis a perfect group. When the series terminates with the identical operation, G is said to be anintegrablegroup; in the contrary case G is callednon-integrable.
If G is an integrable group of order r, the infinitesimal operations X1, X2, ..., Xrwhich generate the group may be chosen so that X1, X2, ..., Xr1, (r1< r) generate the first derived group, X1, X2, ..., Xr2, (r2< r1) the second derived group, and so on. When they are so chosen the constants cijsare clearly such that if rp< i ≤ rp+1, rq< j ≤ rq+1, p ≥ q, then cijsvanishes unless s ≤ rp+1.
In particular the generating operations may be chosen so that cijsvanishes unless s is equal to or less than the smaller of the two numbers i, j; and conversely, if the c’s satisfy these relations, the group is integrable.
A simple group, as already defined, is one which has no self-conjugate subgroup. It is a remarkable fact that the determinationSimple groups.of all distinct types of simple continuous groups has been made, for in the case of discontinuous groups and groups of finite order this is far from being the case. Lie has demonstrated the existence of four great classes of simple groups:—
(i.) The groups simply isomorphic with the general projective group in space of n dimensions. Such a group is defined analytically as the totality of the transformations of the form
where the a’s are parameters. The order of this group is clearly n(n + 2).
(ii.) The groups simply isomorphic with the totality of the projective transformations which transform a non-special linear complex in space of 2n − 1 dimensions with itself. The order of this group is n(2n + 1).
(iii.) and (iv.) The groups simply isomorphic with the totality of the projective transformations which change a quadric of non-vanishing discriminant into itself. These fall into two distinct classes of types according as n is even or odd. In either case the order is ½n(n + 1). The case n = 3 forms an exception in which the corresponding group is not simple. It is also to be noticed that a cyclical group is a simple group, since it has no continuous self-conjugate subgroup distinct from itself.
W. K. J. Killing and E. J. Cartan have separately proved that outside these four great classes there exist only five distinct types of simple groups, whose orders are 14, 52, 78, 133 and 248; thus completing the enumeration of all possible types.
To prevent any misapprehension as to the bearing of these very general results, it is well to point out explicitly that there are no limitations on the parameters of a continuous group as it has been defined above. They are to be regarded as taking in general complex values. If in the finite equations of a continuous group the imaginary symbol does not explicitly occur, the finite equations will usually define a group (in the general sense of the original definition) when both parameters and variables are limited to real values. Such a group is, in a certain sense, a continuous group; and such groups have been considered shortly by Lie (cf. Lie-Engel, iii. 360-392), who calls themrealcontinuous groups. To these real continuous groups the above statement as to the totality of simple groups does not apply; and indeed, in all probability, the number of types ofrealsimple continuous groups admits of no such complete enumeration. The effect of limitation to real transformations may be illustrated by considering the groups of projective transformations which change
x2+ y2+ z2− 1 = 0 and x2+ y2− z2− 1 = 0
respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformation
x′ = x, y′ = y, z′ = z√ (−1),
the general continuous groups which transform the two quadrics respectively into themselves are simply isomorphic. This is not, however, the case for therealcontinuous groups. In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators. The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple.
Among the groups isomorphic with a given continuous group thereThe adjunct group.is one of special importance which is known as theadjunctgroup. This is a homogeneous linear group in a number of variables equal to the order of the group, whose infinitesimal operations are defined by the relations
where cijsare the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves.
That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants. It is thus found that (XpXq) = ΣscpqsXs. The X’s, however, are not necessarily linearly independent. In fact, the sufficient condition that ΣjajXjshould be identically zero is that Σjajcijsshould vanish for all values of i and s. Hence if the equations Σjajcijs= 0 for all values of i and s have r′ linearly independent solutions, only r − r′ of the X’s are linearly independent, and the isomorphism of the two groups is multiple. If Y1, Y2, ..., Yrare the infinitesimal operations of the given group, the equations
Σjajcijs= 0, (s, i = 1, 2, ..., r)
express the condition that the operations of the cyclical group generated by ΣjajYishould be permutable with every operation of the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r′ each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations.
We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made betweenContinuous groups of the line of the plane, and of three-dimensional space.two simply isomorphic groups. We proceed to discuss the classification and theory of groups when their form is regarded as essential; and this is a return to a more geometrical point of view.
It is natural to begin with the projective groups, which are the simplest in form and at the same time are of supreme importance in geometry. The general projective group of the straight line is the group of order three given by
where the parameters are the ratios of a, b, c, d. Since
is an operation of the above form, the group is triply transitive. Every subgroup of order two leaves one point unchanged, and all such subgroups are conjugate. A cyclical subgroup leaves either two distinct points or two coincident points unchanged. A subgroup which either leaves two points unchanged or interchanges them is an example of a “mixed” group.
The analysis of the general projective group must obviously increase very rapidly in complexity, as the dimensions of the space to which it applies increase. This analysis has been completely carried out for the projective group of the plane, with the result of showing that there are thirty distinct types of subgroup. Excluding the general group itself, every one of these leaves either a point, a line, or a conic section unaltered. For space of three dimensions Lie has also carried out a similar investigation, but the results are extremely complicated. One general result of great importance at which Lie arrives in this connexion is that every projective group in space of three dimensions, other than the general group, leaves either a point, a curve, a surface or a linear complex unaltered.
Returning now to the case of a single variable, it can be shown that any finite continuous group in one variable is either cyclical or of order two or three, and that by a suitable transformation any such group may be changed into a projective group.
The genesis of an infinite as distinguished from a finite continuous group may be well illustrated by considering it in the case of a single variable. The infinitesimal operations of the projective group in one variable are d/dx, x(d/dx), x2(d/dx). If these combined with x3(d/dx) betaken as infinitesimal operations from which to generate a continuous group among the infinitesimal operations of the group, there must occur the combinant of x2(d/dx) and x3(d/dx). This is x4(d/dx). The combinant of this and x2(d/dx) is 2x5(d/dx) and so on. Hence xr(d/dx), where r is any positive integer, is an infinitesimal operation of the group. The general infinitesimal operation of the group is therefore ƒ(x)(d/dx), where ƒ(x) is an arbitrary integral function of x.
In the classification of the groups, projective or non-projective of two or more variables, the distinction between primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite family of curves ƒ(x, y) = C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not. In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F(x, y, z) = C, which are interchanged among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) = a, H(x, y, z) = b, which are so interchanged.
In regard to primitive groups Lie has shown that any primitive group of the plane can, by a suitably chosen transformation, be transformed into one of three definite types of projective groups; and that any primitive group of space of three dimensions can be transformed into one of eight definite types, which, however, cannot all be represented as projective groups in three dimensions.
The results which have been arrived at for imprimitive groups in two and three variables do not admit of any such simple statement.
We shall now explain the conception of contact-transformations and groups of contact-transformations. This conception,Contact transformations.like that of continuous groups, owes its origin to Lie.
From a purely analytical point of view a contact-transformation may be defined as a point-transformation in 2n + 1 variables, z, x1, x2, ..., xn, p1, p2, ..., pnwhich leaves unaltered the equation dz − p1dx1− p2dx2− ... − pndxn= 0. Such a definition as this, however, gives no direct clue to the geometrical properties of the transformation, nor does it explain the name given.
In dealing with contact-transformations we shall restrict ourselves to space of two or of three dimensions; and it will be necessary to begin with some purely geometrical considerations. An infinitesimal surface-element in space of three dimensions is completely specified, apart from its size, by its position and orientation. If x, y, z are the co-ordinates of some one point of the element, and if p, q, −1 give the ratios of the direction-cosines of its normal, x, y, z, p, q are five quantities which completely specify the element. There are, therefore, ∞5surface elements in three-dimensional space. The surface-elements of a surface form a system of ∞2elements, for there are ∞2points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of ∞2elements, for there are ∞1points on the curve, and at each ∞1surface-elements containing the tangent to the curve at the point. Similarly the surface-elements which contain a given point clearly form a system of ∞2elements. Now each of these systems of ∞2surface-elements has the property that if (x, y, z, p, q) and (x + dx, y + dy, z + dz, p + dp, q + dq) are consecutive elements from any one of them, then dz − pdx − qdy = 0. In fact, for a system of the first kind dx, dy, dz are proportional to the direction-cosines of a tangent line at a point of the surface, and p, q, −1 are proportional to the direction-cosines of the normal. For a system of the second kind dx, dy, dz are proportional to the direction-cosines of a tangent to the curve, and p, q, −1 give the direction-cosines of the normal to a plane touching the curve; and for a system of the third kind dx, dy, dz are zero. Now the most general way in which a system of ∞2surface-elements can be given is by three independent equations between x, y, z, p and q. If these equations do not contain p, q, they determine one or more (a finite number in any case) points in space, and the system of surface-elements consists of the elements containing these points;i.e.it consists of one or more systems of the third kind.
If the equations are such that two distinct equations independent of p and q can be derived from them, the points of the system of surface-elements lie on a curve. For such a system the equation dz − pdx − qdy = 0 will hold for each two consecutive elements only when the plane of each element touches the curve at its own point.
If the equations are such that only one equation independent of p and q can be derived from them, the points of the system of surface-elements lie on a surface. Again, for such a system the equation dz − pdx − qdy = 0 will hold for each two consecutive elements only when each element touches the surface at its own point. Hence, when all possible systems of ∞2surface-elements in space are considered, the equation dz − pdx − qdy = 0 is characteristic of the three special types in which the elements belong, in the sense explained above, to a point or a curve or a surface.
Let us consider now the geometrical bearing of any transformation x′ = ƒ1(x, y, z, p, q), ..., q′ = ƒ5(x, y, z, p, q), of the five variables. It will interchange the surface-elements of space among themselves, and will change any system of ∞2elements into another system of ∞2elements. A special system,i.e.a system which belongs to a point, curve or surface, will not, however, in general be changed into another special system. The necessary and sufficient condition that a special system should always be changed into a special system is that the equation dz′ − p′dx′ − q′dy′ = 0 should be a consequence of the equation dz − pdx − qdy = 0; or, in other words, that this latter equation should be invariant for the transformation.
When this condition is satisfied the transformation is such as to change the surface-elements of a surface in general into surface-elements of a surface, though in particular cases they may become the surface-elements of a curve or point; and similar statements may be made with respect to a curve or point. The transformation is therefore a veritable geometrical transformation in space of three dimensions. Moreover, two special systems of surface-elements which have an element in common are transformed into two new special systems with an element in common. Hence two curves or surfaces which touch each other are transformed into two new curves or surfaces which touch each other. It is this property which leads to the transformations in question being called contact-transformations. It will be noticed that an ordinary point-transformation is always a contact-transformation, but that a contact-transformation (in space of n dimensions) is not in general a point-transformation (in space of n dimensions), though it may always be regarded as a point-transformation in space of 2n + 1 dimensions. In the analogous theory for space of two dimensions a line-element, defined by (x, y, p), where 1 : p gives the direction-cosines of the line, takes the place of the surface-element; and a transformation of x, y and p which leaves the equation dy − pdx = 0 unchanged transforms the ∞1line-elements, which belong to a curve, into ∞1line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch.
One of the simplest instances of a contact-transformation that can be given is the transformation by reciprocal polars. By this transformation a point P and a plane p passing through it are changed into a plane p′ and a point P′ upon it;i.e.the surface-element defined by P, p is changed into a definite surface-element defined by P′, p′. The totality of surface-elements which belong to a (non-developable) surface is known from geometrical considerations to be changed into the totality which belongs to another (non-developable) surface. On the other hand, the totality of the surface-elements which belong to a curve is changed into another set which belong to a developable. The analytical formulae for this transformation, when the reciprocation is effected with respect to the paraboloid x2+ y2− 2z = 0, are x′ = p, y′ = q, z′ = px + qy − z, p′ = x, q′ = y. That this is, in fact, a contact-transformation is verified directly by noticing that
dz′ − p′dx′ − q′dy′ = −d (z − px − qy) − xdp − ydq = −(dz − pdx − qdy).
A second simple example is that in which every surface-element is displaced, without change of orientation, normal to itself through a constant distance t. The analytical equations in this case are easily found in the form
p′ = q, q′ = q.
That this is a contact-transformation is seen geometrically by noticing that it changes a surface into a parallel surface. Every point is changed by it into a sphere of radius t, and when t is regarded as a parameter the equations define a cyclical group of contact-transformations.
The formal theory of continuous groups of contact-transformations is, of course, in no way distinct from the formal theory of continuous groups in general. On what may be called the geometrical side, the theory of groups of contact-transformations has been developed with very considerable detail in the second volume of Lie-Engel.
To the manifold applications of the theory of continuous groups in various branches of pure and applied mathematicsApplications of the theory of continuous groups.it is impossible here to refer in any detail. It must suffice to indicate a few of them very briefly. In some of the older theories a new point of view is obtained which presents the results in a fresh light, and suggests the natural generalization. As an example, the theory of the invariants of a binary form may be considered.
If in the form ƒ = a0xn+ na1xn−1y + ... + anyn, the variables be subjected to a homogeneous substitution
x′ = αx + βy, y′ = γx + δy,
(i.)
and if the coefficients in the new form be represented by accenting the old coefficients, then
a′0= a0αn+ a1nαn−1γ + ... + anγn,a′1= a0αn−1β + a1{(n−1) αn−2βγ + αn−1δ} + ... + anγn−1δ,· · · · ·a′n= a0βn+ a1nβn−1δ + ... + anδn;
a′0= a0αn+ a1nαn−1γ + ... + anγn,
a′1= a0αn−1β + a1{(n−1) αn−2βγ + αn−1δ} + ... + anγn−1δ,
· · · · ·
a′n= a0βn+ a1nβn−1δ + ... + anδn;
(ii.)
and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which αδ − βγ = 1, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations y(∂/∂x) and x(∂/∂y). Hence withthe same limitations on α, β, γ, δ the totality of the substitutions (ii.) forms a simply isomorphic continuous group of order 3, which is generated by the two infinitesimal transformations
and
The invariants of the binary form,i.e.those functions of the coefficients which are unaltered by all homogeneous substitutions on x, y of determinant unity, are therefore identical with the functions of the coefficients which are invariant for the continuous group generated by the two infinitesimal operations last written. In other words, they are given by the common solutions of the differential equations
Both this result and the method by which it is arrived at are well known, but the point of view by which we pass from the transformation group of the variables to the isomorphic transformation group of the coefficients, and regard the invariants as invariants rather of the group than of the forms, is a new and a fruitful one.
The general theory of curvature of curves and surfaces may in a similar way be regarded as a theory of their invariants for the group of motions. That something more than a mere change of phraseology is here implied will be evident in dealing with minimum curves,i.e.with curves such that at every point of them dx2+ dy2+ dz2= 0. For such curves the ordinary theory of curvature has no meaning, but they nevertheless have invariant properties in regard to the group of motions.
The curvature and torsion of a curve, which are invariant for all transformations by the group of motions, are special instances of what are known asdifferential invariants. If ξ(∂/∂x) + η(∂/∂y) is the general infinitesimal transformation of a group of point-transformations in the plane, and if y1, y2, ... represent the successive differential coefficients of y, the infinitesimal transformation may be written in the extended form
where η1δt, η2δt, ... are the increments of y1, y2, .... By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.
If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. Conversely, if an ordinary, differential equation ƒ(x, y, y1, y2, ...) = 0 admits the transformations of a continuous group,i.e.if the equation is unaltered when x and y undergo any transformation of the group, then ƒ(x, y, y1, y2, ...) or some multiple of it must be a differential invariant of the group. Hence it must be possible to find two independent differential invariants α, β of the group, such that when these are taken as variables the differential equation takes the form F(α, β, dβ/dα, d2β/dα2, ...) = 0. This equation in α, β will be of lower order than the original equation, and in general simpler to deal with. Supposing it solved in the form β = φ(α), where for α, β their values in terms of x, y, y1, y2, ... are written, this new equation, containing arbitrary constants, is necessarily again of lower order than the original equation. The integration of the original equation is thus divided into two steps. This will show how, in the case of an ordinary differential equation, the fact that the equation admits a continuous group of transformations may be taken advantage of for its integration.
The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. It is quite impossible here to give any account of all that Lie and his followers have done in this direction. An entirely new mode of regarding the problem of the integration of a differential equation has been opened up, and in the classification that arises from it all those apparently isolated types of equations which in the older sense are said to be integrable take their proper place. It may, for instance, be mentioned that the question as to whether Monge’s method will apply to the integration of a partial differential equation of the second order is shown to depend on whether or not a contact-transformation can be found which will reduce the equation to either ∂2z/∂x2= 0 or ∂2z/∂x∂y = 0. It is in this direction that further advance in the theory of partial differential equations must be looked for. Lastly, it may be remarked that one of the most thorough discussions of the axioms of geometry hitherto undertaken is founded entirely upon the theory of continuous groups.
Discontinuous Groups.
We go on now to the consideration of discontinuous groups. Although groups of finite order are necessarily contained under this general head, it is convenient for many reasons to deal with them separately, and it will therefore be assumed in the present section that the number of operations in the group is not finite. Many large classes of discontinuous groups have formed the subject of detailed investigation, but a general formal theory of discontinuous groups can hardly be said to exist as yet. It will thus be obvious that in considering discontinuous groups it is necessary to proceed on different lines from those followed with continuous groups, and in fact to deal with the subject almost entirely by way of example.