WILLIAMS. — FINITE CONTINUOUS GROUPS. 



107 



Example IV. 



a X ax par 



e" q, xe" q, . . . , x^e" q, ij q, p 

 K — 1, 2, . . . , W2, «K = const., 2pK + m = r — 2, r>3 



Put p^ =: 1, K := m = 1, and r = 4. We then have the group 



and 



Therefore, 



e 'q, xe'^^q, y q, p ; 

 V = (flj e'^' + a^x e"^ + a^i/) q + 04^3 . 



Ux = 04, ZPa: = 0, . . . ?7"a: = 0, 

 a;' = X + a^ ; 



so that 

 and 



fT'^y = aiOge"^ + a^agxe'"' + a^y + e*^(aia4a + Oo^O + xe"'*a2a4«, 



f7^y = a^a^e'^ -\-a<2,a^xe'^ -{■aiy-{-e'^ {axai^aar^-{-a^a^a^-\-xe'^ a^a^aa^ 

 + e"^ (ai a/ «- + 2 as 04^ a) + a; e»^ Og a^ a% 



U^y — a^aie'^^a^aixe'^-{-a^^y-\-e'^{axa^aa^-^aia^a^)^xe'^a^a^ua^" 

 + a;e'"a2a4^a^a3, 



Hence 



y' =ye''^ ■\- X e 



Cfn 



(e««* _ e%) 



+ ea2 



(^ a 04 — «3 J 



}fll(a3-aa4)+«2fl4l(e"8-l)-ai(a3-aff4)(e°"«-l)-q2fl'«e°^*(Q3-g"4 + ^] 



(a3-aa4)'-2 



For the values of r, p^ , and k, chosen, this group is continuous. 



