34 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3 d Ser. 



If a 2 n -\- a 2 n = o, and therefore a n ^ o, this is accom- 

 plished by the values 



^ . — a ki „ _ . — a ki 



fi= -, v= i. 



2 a u 2a n 



If a 2 n + a 2 jX ^ o, we derive the equivalent condition 



« 2 u 2 n + a2 ji) = -{? i a \i + « 2 n) + v a n « k i T + 

 v 2 ^(a*, + a 2 n -f- a 2 kl ), 



which, by the generalization of the theorem of Jordan, p. 

 159, has solutions fi, v except when a 2 u -\- a 2 it -\- a 2 kl = o, for 

 which case the condition may be written 



« 2 ii = — 0«ki— va iiY- 

 If — 1 is a square, this has solutions; or we may apply § 7 

 since a 2 a + a 2 k i= — a \> a square. 



9. The transformed of S a by ijkl gives ^a, where 

 a il = Xa il -f ^aji + ^a k i + °"«ii 5 etc. 



If a 2 u + a 2 ji + «\i = #, the values 



— a, 1 a,. 1 a, , — a,, a,, 



X = - _!! , p = JlI-! 1 , v = _|L_L2 , <r = 1 



make a'j x — o , X 2 -f- ^t 2 -J- u 2 -f cr 2 == 1 . 



10. Our invariant subgroup / of H was shown to con- 

 tain the substitution S=S a C\ C 2 not the identity. Apply- 

 ing §§7-9, we transform S successively by O i3i5 (t' = m, 

 m—i, . . 6) or else by O i3i5 T l2 C x when the former does 

 not belong to the group H. We thus obtain in / a substi- 

 tution S\ in which a ml — a m _ 11 = = a 61 = o and also 



d 5 i = o. The latter follows by § 8 since d 2 n -J- a 2 2i = a\\ + « 2 2i 

 differs from 1, whence a 2 51 -|- a 2 4i -\- a 2 31 ^ o. 



11. Next transform ^i successively by j567 (J=m, . . .8), 

 giving a substitution S'=SoCi C 2 leaving £s , • • £ m invari- 

 ant. If /3 52 , /3 62 , /3 72 all differ from zero, we may by § 8 

 suppose /8 72 = <? unless /3 2 52 -j- /3 2 6 2 + /3 2 72 = 0, with — 1 a not- 

 square. In the latter case, we transform S' by #34567 and 

 require that (since /3 51 = Ai = Ai = 0) : 



/3' 72 = X /3 32 + ^ /3 42 + v A2 -l- o- £ 62 -f /> £ 72 = 0, 

 /3' 7 i=X/3 31 -|- ^£ 41 =0. 



