and Homoid Products of Sums qfn Squares. 4-99 



of which sets no one can be identical with /x^' 2 , the sum of the 

 squares of the constituents of Q' fl// , any other of the 480 above- 

 mentioned forms; because w lP and the terms of a n b n . .g u 

 which contain m and Bfy will vary through all these sets, while 

 they are invariable through all these forms. If, in the con- 

 stituents of Q fl// the first here mentioned, we change the signs 

 either of all of iva . . g, or of all of , w l a l .. .(/,, we shall sim- 

 ply reverse the signs of all the roots in /x /y 2 $ if we change the 

 signs of the eight wabcVtfLbfm indicated by any triplet a b c 

 in the system of seven, or if we change the signs of the eight 

 bcdebf'dfp indicated by any two triplets a Q b c o , a d e oi ha- 

 ving any common imaginary, we shall merely reverse the signs 

 of certain of those roots. Finally, after making all these va- 

 riations of sign of the constituents of Q and Q a/ , we may essen- 

 tially alter Q a and Q a ,by exchanging in either or in both of 

 them any other constituent, as c or c p for the real term w or 

 iv p and thus give rise to another series of variations of the 

 eight squares > (X) i fa l f. . . g*. 



Making now a slight change in our notation, for clearness 

 sake, let 



Q^Q/vQ/',...., Q_ i5 Q'_ t , Q"_ f , 



represent any pluquaternions of/ imaginaries, the second set 

 in order differing from the first only in the signs of all the i. 

 Let (51, represent a pluquaternion which arises in the product 

 of two or more, of i imaginaries, or of the order i. And let 



Q i Q' i = <a j + ^, 



3&,- being the function R m , of the second degree, already found. 



2m=:a+l+b+l -fc+1 + . . . +n+ l> 



to n terms, when a > b, b^c, &c. . . , and if ju, 2 ju, 2 beany two 

 sums each of 2m squares ; then 



i* 2 =Q a Q- a +Q 6 Q_ 6 + Q.Q-. + ... +Q»Q-», 



^ 2 =Q' a Q'_ a + Q' 6 Q'_. 6 + Q' c Q'_ c + ... +Q'„Q'_ M ; 



ptff* [Q a Q- a Q' a Q'- a =Q u Q'aQ'- a Q-J =(*.+».)(*-.--».)+ 



[Q.Q-.Q'»Q , -»«Q.Q'*Q , -»Q-.=](^ , .+»'J(«t'„.-»'.) + 



+ 



[Q 6 Q_ 4 Q' a Q^ a =Q 4 Q' a QL a Q_ 4 =](^^ 

 [Q 6 Q_ ft Q' A Q'_ 6 =Q 6 Q' 6 Q'_ 6 Q_,=](e 6 +m A )«a_ i -ia 6 )+ 

 [Q A Q_ A Q' C Q'_ C =Q 6 Q' C Q'_ C Q_ 6 =] (fc',+»',)(*'_,-*' 6 ) + 



&c. &c. ; 



