102 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

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

 curves of our group are given by the equation 



dx dy 



az + a^x «! + a^x -\- a^ay' 



the solution of which gives 



y = a/(i-«) ^"^ + "^^ ^ ^<— + ^(^^ + «*^)''' 



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

 infinitesimal transformation U we put a^ — 0, as — 0, and a^ and a™ finite, 

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

 is Ui = (oi + 02 x) q, and whose path curves are x — const. ; which is 

 then the one-term group associated with the singular transformations T. 



Example II. 



Put r = 3, K = m = 1, and p^ — 1. We then have the group 



e"'^^, xe-"^ q, p. 



(«to) 



The symbol of the general infinitesimal transformation is 



7-7 / a.x \ aX\ I _ «-. 



U = («i e -j- a^x e ) q + a^ j). 

 Hence, 



Ux =^ Oj, 



U-'x = 0, 

 and therefore x' = x + a^. Further, 



TT aX \ (XX 



U y = 016 -\- a<txe , 



TT-'* a X I a X I _ aX 



(J-y =^ aia.,ae -\- ana-^e -\- a.^a^axe , 



T7"a 'i ^ o-X I Ck Q a X I 09 a3? 



T-TW + 1 'iwaaji 'in. — lax, n n ax 



LI ^ y ^ a^a^ a e ■\- na^a^ a. e -{- o^a^ a x e . 



