LINEAR HOMOGENEOUS GROUP IN THE 0JF[2] etc. 207 



Hence 



upon applying 209) and 76). Hence I/} I does not satisfy 209), 

 while MiMjI. does. 



b) If the product N m j >x l. be expressed in the form 210), we have 



<x'rs = u rs , fi rs = p rs (V, s = 1, . . ., m) 



y'rs = rr,, d'rs = #r* (V, s = 1, . . ., m] s =%=m,j) 



Vrm= rrm-i- xa rj , y rj = y rj -f- xa rm + ^K 2 cc rj (r = 1, . . ., m) 



Hence J(a' ; /3 f , y f ; d') equals 



r, s =!,..., 



r=l 



(8 mm + K 



/ m \ 



\r=l / 



But the last two sums are zero by 76) and 193). 



c) An analogous proof holds for N,-j, x (i,j<m), the above 

 terms involving A% 2 not being present. 



d) Since the substitutions Q;j,%, -R/,^,* (i, j = 1, . . ., m) and P iit 

 T i)X (i,j < m, if A = A') may be expressed as a product of the JY/,/ |X 

 and an even number of the M; ( 202), the products jT,- )X Z, /,.?, *Z, 

 etc., will satisfy 209) if Z does. 



Inversely, every substitution S of 6a which satisfies 209) belongs 

 to Ji. In fact, by the proof given in 202, S is of one of the two 

 forms K, KM m , where K is derived from 1 ) M- t M^ Qi,j, x , $*,/, x> 



7? fa A 1 AM\. p.. 7*. /V ,,* <^ wi if 3 2'"\ SI'TIPP # aiioll 



Xtj x \y) j 9 * * 'J )l tji -*-tx \ i J ^ /" kJiiiUc O blldil 



satisfy 209), it is not of the form KM m . It remains to show that 

 these substitutions M { M h Qi,j, K , . . ., T/, x belong to Ji. 



For m > 2, Ji contains Qi,j, x , the transformed of JVJ,^ |X by 

 J^Jf^ (^H=^ j); a l go -Bf,/,x an( l ^-,f,x, the transformed of N itil1t and 

 C,y,x respectively by MiMj. Applying the formulae at the beginning 

 of 202, we reach P i$ and T iilt Tj lfl (i,j<m 9 if A = A'). Then Ji 

 contains Ti }fl lj ftl i, the transformed of the latter by M m Mj. The 

 product of the two gives T^*. 



1) By 196), L and therefore every Gf%f is derived from M^ Jf 2 , Q 2 x and 



