86 



is said to be a point transformation, carrying point x, y into point xi, yi. 

 Here it is assumed that inversely 



X = <i)i (xi, yi), y = ^1 (xi, yi) 



carries the point from xi, yi back to x, y. A point transformation may be 

 looked upon either as a transference of axes from one system to another, 

 not necessarily the same kind of system, or it may be considered as an 

 actual transference of one point into another position in the plane, the 

 axes of reference remaining unchanged. 



2. Group of Transformations. A point transformation containing one 

 or more parameters 



xi = 4> (X, y, a, b, c, . . . k), 

 yi =*(x, y, a, b, c, ... k), 



such that far ao, bo, Co, ko, the point x, y transforms into itself, is said 



to constitute a group of transformations when a succession of two such 



operations may be replaced by one of the same species. That is, if 



X2 = *(xi, yi, ai, bi, ci . , . ) =<1>| 4> (x, y, a, . . .k), ■^ (x, y, a, . . . k), ai. . . ki | 



= O) (X, y, a.2, b,, C2, ... kg), 

 y2 ='!' (x, y, a2, b2, Cj, ... kg), 



where a2 := fi (a, b, . . . k, ai, bi, c, . . . ki), bj = f 2 (a, b, . . ki) , then 



XI = * (x, y, a, b, ... k), yi = t (x, y, a, b, . . . k) 



are the transformations of an r-parameter group, the parameters a, b, c, 

 . . . k being r in number and independent. A similar definition may be 

 given to a group in one, three, four, or n variables*. 



3. The Infinitesimal Transformations. An infinitesimal transformation 

 is defined analytically by 



Sx = f (x, y) <h, (5y — // (X, y) <5t. 



Such a transformation attaches to any point x, y an infinitesimal motion 

 whose projections on the x — , and y — axes are respectively, ^('t and //(H, 

 and whose distance is / f ■^ 4- //^ (h. Lie shows sucli infinitesimal trans- 

 formations to belong to a single-parameter group. 



XI = -I' (x, y, a), yi = * (x, y, a). 



This may be easily seen by letting o,, be the value of a wliich leaves x, y 

 fixed; then 



xi = <I' (x, y, a„ + '5a). y, = I' (x, y, a,> -j- '''a) 



•See Lie — Schoffers, Differential-gleichungen, pp. 24-25. 



