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



Since (o*'%\= Q*'~]j~^, we have 



= O x,tS .O x ^. Q X, ^_Q X, % O x, @ = // 6>^'^ 



Our product will finally take the form 



k" <9 A%/3 = A" 6> A%/9 6>* ,/? . O x, P = h O x,t3 . 

 r,s r,s 2,1 1,2 1,2 



3. Theorem. For «z>4 the maximal invariant sub- 

 group I of H\ is of order 1 or 2 according as m is odd or 

 even. 



The proof is quite similar to that given for the case 

 fl n = 8/ ± 3 in the paper cited. The lemma of § 6 is now 

 replaced by the following 



Lemma. If I contains C i Cj it coincides with H\. 



Thus O x ^ O x,fS transforms d C 3 into Q X ^ G C 3 . 



2,4 1,2 ^1,2 



x (3 

 Hence I contains every Q". . and in particular T 23 C 3 



1 >J 

 (2 being a square). Thus / contains 



(7* C 3 ) {O l2 Ou) {T 23 C3)- 1 {On O l4 )~ l = O l3 2l . 



(T 23 C 3 Tu C 4 ) {On Ou) {T 23 C 3 T n C4)- 1 {O l2 On)' 1 = 



o i3 o 21 . 



In §10 replace O i3i5 T 12 C\ by O m5 O ' . The develop- 



ment at the end of § 18 is unnecessary when 2 is a square. 

 Thus from 



pi + pi -f Pi = 0, Pi 2 + P2 2 = 2, 



it follows that pi = — 2 ; while — 1 was supposed to be a 

 not-square. 



4. Under the subdivision f = 8 / ± 3, the cases fi n = 3 

 and 5 were excluded in the earlier paper. 



Consider the case f" = 5, ra>4- [See p. 41 of paper 

 cited.] Zi 2 C\ extends H to the total group G; indeed, it 

 transforms R into C 2 C 3 R T\ 2 T\ 3 C 2 C 3 , viz. : 



