WILLIAMS. — FINITE CONTINUOUS GROUPS. 105 



The singular transformations 7 1 defined by equations (7) leave invariant, 

 as a whole, the system of lines x = const., but change each line into some 

 other line of the system. Associated with T is a one-term group whose 

 path curves, x = c, are as a whole unchanged by T. The path curves 

 generated by the general infinitesimal transformation of this group are 

 defined by the equation 



dx dy 



a 3 e aX (a t + a 2 x) * 



the solution of which gives 



e aX (ax a + a 2 a x — cto) = « 3 a 2 y + c. (c = const.) 



If, now, in the symbol of the general infinitesimal transformation U, we 

 put a 3 = 0, a 2 = 0, and a x finite, we get the one-term group, whose symbol 

 of infinitesimal transformation is U Y == a l e ax q, and whose path curves 

 are x = const. ; which is then the one-term group associated with the 

 singular transformation T. 



■&■ 



The following groups do not contain singular transformations, and are 

 properly continuous groups. 



Put r = 3 ; we then have the group 



q, <f>(x)q, yq; 



and 

 Therefore, 



U=z a r q + a 2 (j) (x) q + a^yq. 



Ux = 0; 



Uy = «! + « 2 (*) + « :i y< 

 U 2 y = a 1 a 3 + a 2 a 3 <£ (ar) + a 3 2 y, 



U n y = a x o z " ~ i + a 2 a 8 »- i (ar) + a 8 "y. 

 Hence the transformation 7^ of this group is defined by 



