the 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
| a0 | ∂ | + 2a1 | ∂ | + 3a1 | ∂ | + . . . + nan − 1 | ∂ | , |
| ∂a1 | ∂a2 | ∂a3 | ∂an |
and
| na1 | ∂ | + (n − 1)a2 | ∂ | + (n − 2)a3 | ∂ | + . . . + au | ∂ | . |
| ∂a0 | ∂a1 | ∂a2 | ∂au−1 |
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
| a0 | ∂ƒ | + 2a1 | ∂ƒ | + 3a2 | ∂ƒ | + . . . = 0, |
| ∂a1 | ∂a1 | ∂a2 |
| na1 | ∂ƒ | + (n − 1)a2 | ∂ƒ | + (n − 2)a3 | ∂ƒ | + . . . = 0. |
| ∂a0 | ∂a1 | ∂a2 |
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 as differential 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
| ξ | ∂ | + η | ∂ | + η1 | ∂ | + η2 | ∂ | + . . . |
| ∂x | ∂y | ∂y1 | ∂y2 |
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.
The consideration of a discontinuous group as arising from a set of independent generating operations suggests a purely abstract point of view in which any two simply isomorphic groups are indistinguishable. The number of generating operations may be either finite or infinite, but the former case aloneGenerating operations. will be here considered. Suppose then that S1, S2, . . ., Sn is a set of independent operations from which a group G is generated. The general operation of the group will be represented by the symbol SαaSβb . . . Sδd , or Σ, where a, b, . . ., d are chosen from 1, 2, . . ., n, and α, β, . . ., δ are any positive or negative integers. It may be assumed that no two successive suffixes in Σ are the same, for if b = a, then SαaSβb may be replaced by Sα+βa. If there are no relations connecting the generating operations and the identical operation, every distinct symbol Σ represents a distinct operation of the group. For if Σ = Σ1, or SαaSβb . . . Sδd = Sα1a1Sβ1b1 . . . Sδ1d1, then S−δ1d1 . . . S−β1b1S−α1a1SαaSβb . . . Sδd = 1; and unless a = a1, b = b1, . . ., α = α1, β = β1, . . ., this is a relation connecting the generating operations.
Suppose now that T1, T2, . . . are operations of G, and that H is that self-conjugate subgroup of G which is generated by T1, T2, . . . and the operations conjugate to them. Then, of the operations that can be formed from S1, S2, . . ., Sn, the set ΣH, and no others, reduce to the same operation Σ when the conditions T1 = 1, T2 = 1, . . . are satisfied by the generating operations. Hence the group which is generated by the given operations, when subjected to the conditions just written, is simply isomorphic with the factor-group G/H. Moreover, this is obviously true even when the conditions are such that the generating operations are no longer independent. Hence any discontinuous group may be defined abstractly, that is, in regard to the laws of combination of its operations apart from their actual form, by a set of generating operations and a system of relations connecting them. Conversely, when such a set of operations and system of relations are given arbitrarily they define in abstract form a single discontinuous group. It may, of course, happen that the group so defined is a group of finite order, or that it reduces to the identical operation only; but in regard to the general statement these will be particular and exceptional cases.
An operation of a discontinuous group must necessarily be specified analytically by a system of equations of the form
x′s = ƒs (x1, x2, . . ., xn; a1, a2, . . ., ar), (s = 1, 2, . . ., n),
and the different operations of the group will be given by different sets of values of the parameters a1, a2, . . ., ar. No one of these parameters is susceptible of continuous variations, but at least one must be capable of taking a number of values which is not finite, if the group is not one Properly and improperly discontinuous groups.of finite order. Among the sets of values of the parameters there must be one which gives the identical transformation. No other transformation makes each of the differences x′1 − x1, x′2 − x2, . . ., x′n − xn vanish. Let d be an arbitrary assigned positive quantity. Then if a transformation of the group can be found such that the modulus of each of these differences is less than d when the variables have arbitrary values within an assigned range of variation, however small d may be chosen, the group is said to be improperly discontinuous. In the contrary case the group is called properly discontinuous. The range within which the variables are allowed to vary may clearly affect the question whether a given group is properly or improperly discontinuous. For instance, the group
