96 CHAPTER II. 



Writing ,-, ^ for /, ^- in S 2 , and applying conditions 79) , viz., 

 Jf?84=l, E li =E 2 ^=0 ) R li =B* i =E* i =Q (j = 5, 6, . . ., 2m), 

 we find that $ takes tlie form 



the indices /, rji (i = 3, . . ., m), not being altered by S 2 and the 

 substitutions below, are not written in the formulae. 

 The group J contains the product 



where is a linear function of 17 i^ 17 %. 



a) Suppose first that $ 3 is not the identity. If 1 a n =[= 0, we 

 may define r by the equation 



Then e7 contains S = L\^ 1 S$L\^, which has the form 



iia> u / , QN 



ia - ^2 + ffaiii + fti^i + ^22^2. ^2 - ^2- 



Applying the conditions E 13 == -^3= of 79), we find that y lg 

 21 == 0, so that $ 4 has the following form (with a =f= 0): 



If, on the contrary, 1 n = 0, J will contain M 1 1 S B M 1J which 

 is not the identity and has the form 81). In either case, J contains 

 a substitution 81) in which a and /3 are not both zero. 



If a = 0, 4= 0, 81) is of the form L it p 4= J. If a + 0, 

 7 contains the transformed of 81) by $3, i, Aj gi v i n g ^ ne substitution 



Taking A = /3/ 2 a, this becomes N^^ a - Then J contains 

 82) i,, _ , = JVi, 2 , M 7 J JTiTi " M t (M t I*, j)- 1 JV,, ,. (M,L 2 , ,). 



Transforming by P 12 , we reach Z 2 , _2. In either case, J" contains 

 a substitution of the form jC 2 , ;. (k =j= ^)- 



We next prove that J contains all the generators Lf tfl) M t and 

 Ni t j tft of the group G. Having L^i, J contains the product 



2V,* 1 I'M 2*, = !,* (T any mark 4= 0). 



