L02 DINGS OF THE AMERICAN ACADEMY. 



some other line of th m. A \ with T is a one-term group 



whose path carves, x c, are as a whole unchanged by /'. The path 

 curves oi our group are given by the equation 



dx dy 



a^ + a^x a x + a a x + a^uy' 



the solution of which gives 



" •• , N "i "i - " " . 



y = V(i-«) (a8 + g '* ) - T* y(«» + a «*) a . 



where y is the constant of integration. If in the symbol of the general 

 infinitesimal transformation D we put a 4 0, ". 0, and ay and a* finite, 

 we get the one-term group whose symbol of infinitesimal transformation 



i- f\ = (ai + "■:■')'/. and whose path curves are x = const.; which is 

 thru the one-term group associated with the singular transformations /'. 



Example II. 



a K x p K a K i 



e (7, X e q, . . . x <■ y. j> 

 k=1, 2, . . . m, u K = const.. %p„ + /// /■ — 1, 7- > 2. 

 f 



Put r = 3, <c = m = 1, and ^ = 1. We then have the group 



a. X 



e (j. xe y, p. 

 («*0) 



The symbol of the general infinitesimal transformation is 



Z7 = (a, e ax + a, x e aT ) 7 + a, ;j. 

 Hence, 



. 'r = r/ g , 



/ .r = 0, 



/ • ■ .r = 0, 

 and therefore x = x + a 3 . Further, 

 Uy = a l e ax + a s xe ax i 



tr.i ax 1 a x 1 a .r 



-_y = a a a cu ■' a .'■ + n. a a x <^ . 



/ - 1 11 " a ,r 1 I* n — 1 _o(B 1 n n 



I y = n x ii., u e •+- Bfl.flj a « + "■_• " ' c . 



