WILLIAMS. — FINITE CONTINUOUS GROUPS. 



105 



The singular transformations T 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 Z" is a one-term group whose 

 path curves, a: = 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 



the solution of which gives 



e"'^ ipi tt -{■ cu a X — a.2) = as a" y + c. 



(c = const.) 



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

 put ttg = 0, cTo = 0, and Oj finite, we get the one-term group, whose symbol 

 of infinitesimal transformation is l/i = aie'^^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 



7, <A(x)(7, yq-; 



and 

 Therefore, 



U z=^a^q ^ a.,(f> (x) q -f- a.yq. 



Ux = 0', 



Uy = ai + a.2 <j!) (x) + a.y, 



IPy — a^ «3 + «2 "z 4> (^) + «3^y, 



Vy^ «i ffs"- ^ + «2 «3" - 1 </> (x) + a,-y. 

 Hence the transformation T^ of this group is defined by 



