52 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



Vi = #11 Vi + *A (^13 — i ^12) V2 — Yi (#13 + * #12) tj 3 , 



772' = (#31 + * ^2l) Vl -\- % (#22 l #32 + 2 #23 + #33) V2 + 



}4 (#22 Z #32 t #23 #33) V3 5 



773' = ( #31 -f ?'#2l) Vi -\~ /4 (#22 + I #32 + * ^23 ^33) ^2 + 



Yz (#22 + *#32 «#23 + ^33) *?3  



It is proven below that the coefficient 



]/ 2 (# 22 -f #33) + t/2 (#23 #32) = # 2 , 



# being a complex of the form p -\- <r i, where p and a are 

 marks of the G F [_^ n ] . It follows that the coefficient of 

 773 in 772' is a square, viz. : 



y 2 (#22 #33) if 2 (#23 -f- #32) = /3 2 . 



In fact, by virtue of the orthogonal relations (2), p. 30, 



# 2 # 2 = % (#22 2 + #23 2 *32 2 #33 2 ) «/4 (2 # 22 #32 + 2 *23 ^3) 



= % (#31 + *'#2l) 2 . 



The coefficients of t/ 2 and i]s in ^3' are squares, say <y 2 and 

 S 2 , since they are the conjugates of /3 2 and # 2 respectively. 

 We may verify that 



# 2 7 2 = yi (^13 — t #i2) 2 , 



whose conjugate is therefore /3 2 S 2 . Further, we find 1 



# 2 /3 2 



7 2 S 2 



#22 #33 ^32 #23 '— #11. 



Also (# § -j- /3 7) 2 when expanded reduced to #n 2 . 

 Hence x S — 7 = 1 . 



Our substitution ^1 thus becomes 2 



r x 8 -f /3 7 #7 ^ 8 



2 # /3 

 2 7 8 



#* 



/3 2 

 S 2 j 



[*« — £7 = 1] 



Giving aSi the notation ' ^ we verify the composition 

 formula for Si' Si : 



1 Baltzer, Determinanten, § 14, 5, page 189. 



2 Compare Fricke-Klein, Automorphe Functionen, I, p. 14; also Weber, 

 Algebra, II, p. iyo. 



