and Homoid Products of Sums ofn Squares. 507 



considered for convenience of reduction (when possible) as 

 fyfi' c 3 c 4g6 e 7> * s arbitrary, whatever multiplet A may be, by an 

 indeterminateness like that which we have met with in con- 

 structing a system of seven triplets. Let us leave it undeter- 

 mined thus : 



+ b • A = (bf x • c 3 c 4 g& 7 = -cg x ' c 3 c 4 gg 7 m ) ± cg x c 3 c^g^ v 



because b*fy = — c K g& ; and we do not stay here to ask 

 whether the transformation of cg x 'c 3 c 4 g 6 e 7 i nto c g\ c s c 4g6 e 7 m " 

 volves a change of sign or not, We suppose that the question 

 is capable of being congruously decided, either in one way or 

 in two ways. 



±c'A=(cf x 'c 3 c 4 g^ 7 =zbg l -c 3 c 4 g^e 7 = )±bg x c 3 c 4 g 6 e v 



because b a g^ ^c^f^. The object is here to show, that there 

 is no necessity that b, d, or f with any subindex, should ap- 

 pear except in the first place of the resulting irreducible mul- 

 tiplet, when the subindices are arranged in ascending order. 



± h A = {b l f l -c 3 c 4 g & e 7 -d 1 'c 3 c 4 g 6 e 7 = ) ± d^c^^ 



± C l A = i C lfl' C 3p4g6 e 7= - e l' C 3 C 4g6 e 7 = ) ± WtftgePr 



+ £ 2 A m {bj f l 'c 3 c 4 g 6 e 7 =g l csc 3 c 4 g 6 e 7 = )±g l c< l c 3 c 4 g 6 e 7i 

 ± c 2 A = {pJx'Cf&fr = -/ x c 2 - c 3 c 4g6 e 7 = ) ±f l c^ s fi A g^ 7t 

 ± &j A = {hc 3 'c 4 g 6 e 7 f x = a • c 4 g 6 e 7 f x = ) ± b 4 -g 6 e 7 f x , 



= {±fl b 4g(f7= +g\ C 4g^7 = ) ±glC4g&7> 

 ± C 3 A = ( C 3<V C 4g6 e 7fl = - C 4g6 e 7fl fa ) ±f\ C 4g(?T 



For b x f x = d x ; c x f x =-e x ; b 9 f x =g x c t \ b 3 c 3 =a-, ac 4 =-b 4 ; 

 f x b 4 =—g x c 4 . The equation + b 3 A = +g\C 4 gge 7 is meant to 

 assert that 6 3 A 3 , with one sign or other, is equivalent to the 

 quadruplet g\C 4 g<.e T 



Enough is here done to show, that, on our suppositions, 

 the operation Q.A will give rise to terms affected with sex- 

 tuplet, quintuplet, or quadruplet imaginaries, each reducible, 

 with one or the other sign, to an equivalent, in which the sub- 

 indices shall ascend, and which shall exhibit b, dor fin no 

 place except the first, and in the other places, no letters ex- 

 cept repetitions of c, e,-g. If A were a quintuplet having c, e, 

 or g, in the first place, instead of/, it would be easy to show 

 that the multiplication Q f A would yield results of the same 

 forms. Generally, if V OT be the function of £-plet imaginaries, 

 the operation Q { V m will produce all the (e+l)plets of the 

 system which have (£+1) different subindices, together with 

 a number of f-plets and of (e— 1 )plets, the whole of which can 

 be reduced to equivalents which shall exhibit b, d or/ only 



2 L 2 



