and Homoid Products of Sums of n Squares. 503 



because 



Q.R =R Q .: 



wherefore 



— (b 'n p )(b 'ti p o )=n p 'b o 'b ' n p o ~ —n p -n p 



or 



-S S = + B 2 +B?+B 2 +... + C»+&c. 



mm ' ip ' Ip ' rip ' tp ' 



+ certain terms containing imaginaries of the sixth degree. 



It is too evident to require proof that these latter terms in 

 — S m S OT , as well as the imaginary terms of the fifth degree in 



— )\ m O m — O w -tVjre) 



and those of the fourth degree in 



must respectively in pairs destroy each other; wherefore, 

 whatever be Q.'", 



Qi"S m = S m Q'". i ; 

 or, if 



whose modulus is SL, 

 Further, 



Q_ ( -a'_ 1 Q"_,--Q"'_ li =(0i_ i a'"_ i -^ TO a"'- i 



when T m is supposed to be reduced to as few terms as pos- 

 sible, the number of which we do not yet know, any more than 

 that of the terms in iH, OT = R m + S m . 



Q!" Q/' Q/ Q.- Q_,Q'_,Q"_,Q'"_, = Q.'" Q ." Q i y 9 Q'_,Q"_iQ"'_, 



= Q!"Q!'Q!L i ixyQ!"_ i = qMuH?i ftW'^ 



••• f*VW 8 =*^-<-^- (»» + T w ) + (% + T m )(ft_, 



-(m m + T m ,) 2 - 

 =m 2 -Ol < .T w + T m ^. t -U m fX m -U m T m -T„M m -T m T ra . 

 By reasoning as before it can be shown, that the only real 



