LINEAR HOMOGENEOUS GROUP IN THE (7F[2] etc. 203 



c) If iy= yij = (j = 1, . . ., ~k 1), but ax* and yi* not both 

 zero, we may, for fc < w, proceed as in case a) or b) and obtain a 

 substitution T' which replaces it by and is derived from the sub- 

 stitutions 195). We then take T = T'Pi*. 



d) If KIJ= y\j= (j = 1, . . ., m 1), the proof given in c) 

 applies if Z = 0, since then P lm is generated by the substitutions 195) 

 of ft. For A = A f , this case cannot exist, since the equation 



aimyim+ *'! + AVim= 



requires aim=yim=0 on account of the irreducibility of (> Then 

 would / t = 0. 



It follows that S = TS 1 , where S leaves | fixed but is a sub- 

 stitution belonging to ft. Let S t replace % by 



where, by 78), ^ = 1, and 194), 



m 



198) d n A lft/^4 A/8m+ Aa? m = 0. 



The product 



replaces |j by |j and ^ by 



(ft 2 ^12 +" + ftm^lm + A]8! m + AW^fe + % -j- OJi^S; + 



which equals f since the coefficient of | t equals ftj by 198). 



We may therefore set $ 1 =$'$ 2 , where $ 2 is a substitution of 

 ft which leaves ^ and ^ fixed. Then by 78), 



a = fti = y<i = fti = (i = 2, . . ., m). 



The relations holding between a^-, /3,^-, y^-, d^- (, j = 2, . . ., m) are 

 seen to be the relations 78) and 194) when m 1 is written for m. 

 Proceeding with $ 2 as we did with S, etc., we find ultimately that 

 = T'I, where T' is derived from the substitutions 195), while I 

 is a substitution of ft which affects only | m and iy m , 



The conditions 78), 193) and 194) become, for m = 1, 



199) ad + 0;/ = l, a/J + Aa 2 -|- A/3 2 = A, yd + Ay 2 + Ad 



200) d/3 + Ad 2 -f A/3 2 =>1, ya + Ay 2 -f 



Combining 199) with 200), we may replace 200) by 



201) ft(a + d) = y(a + d) = A( + d) 2 . 



