WILLIAMS. — FINITE CONTINUOUS GROUPS. 99 



a/'^/' + rV-i). 



(2) *■ 



. bnb 3 a b 6 Jj „ J 



u 4 (« — 1) a o 4 



Substituting in (2) the values of x' and y' from (1), we get the transfor- 

 mation T b T a which carries the point P into the point P" . The equa- 

 tions defining T b T a are then 



x " = x e «* + b * + a Ji (J* + & < _ M + *? ( /, _ 1} 



(3) "* bi 



y" = ye« ( ' h + *> + -^- J ^ (e° (ai + * j) - e* + ' * 4 ) + £ («' '* - e" 1 ) I 

 ct — 1 / (-l± t> 4 i 



. <h a 3 f «(«4+ 6 4) « 4 +a6 4 ab t ab t . «l , a(a t +b t ) ab 4 



-\ —, 7-Ae — ae —e + ae ) -j (e — e ) 



«o 4 "( i « — 1) m# 4 



+ -— -(e - a e -1 + a) + —r(e - 1). 



a 4 (_« — 1 J m o 4 



If now 7^ 7^„ which is also a transformation of our group, is equiva- 

 lent to a transformation T c , generated by the infinitesimal transformation 

 (e, + CoX + c i uy)q + (c 8 + c A x)p, — that is, if T b T a — T c , we have 

 also 



3 /_ c 4 



// ^4 I '» / ^4 1 \ 



x = xe H (e — 1), 



C A 



(4) 



+ -£L (e ^_i). 



« c 4 



Whence, first, 



and therefore, 



e* = e* + \ 



«C ( a («4 + ^4) 



e = e 



c 4 = a 4 + h + 2 k 7r t, 



«c 4 = a (« 4 + A 4 ) + 2 K ' *" *> 



