and Homoid Products of Sums of n Squares. 497 



the imaginaries must of necessity destroy each other, and that 

 in every term. From this it follows, that the duad imagina- 

 ries of the condition-function possess the properties 



(Mo) 2 =(4>/o) 2 =..=-l> 



b n Q ' d Q l + d l ' b o n=0, 

 c o 'b o n o + b o n -c o =0. 

 And if it can be proved that 



c 'b n = b -n c 

 in all cases, it will follow, that whenever b Q ' n Q c u is imagina?y r 

 b o 'n o c o + b o n o 'c o =0. 



By equations (G.), for any two pluquaternions of (2n— 1) 

 imaginaries, 



QaM-a+QaQ-a^A + B) 



= 2(ww, + aa t + . . . + rr t ) ; 

 wherefore, Q a Q a , being (Q B/I +R), 



Qa / Q-« // +Q^ / Q-a / =2(w i w w + a / a // + fl i fl H + ... +r,r n ). 

 But, by equations (F.), 



Q„,Q-«„ + Qa„Q-a =Q„,Q- a/ Q-a + Q«Q«, Q-a, + Q«, R- RQ- fl/ 

 = (Q-a+Qa)Q U/ Q- a/ , 



since Q a/ Q_ a/ is real; 



= 2te*/* i 2 ; 

 therefore 



w=(iD l vo ll + ap u + . . . +r/*j • (w i 2 + « y 2 + . . . + r, 2 )- 1 ; 



and in the same way can be proved 



v> l ={ww ll +aa ll + • • • + rr / ;)'(a£ + a 9 + . . . +r 2 )- 1 . 



Let a Q a =Q' a , and a Q Q a// =Q' fl/ : Q' a and Q'„„ are plainly 

 pluquaternions, whose real terms are — a and — a lt instead of 

 tso and w u . We can therefore deduce by the preceding rea- 

 soning, from the equation Q' a Q„, . = Q' a „ + a R, the value of a 

 in terms of w,a y . . r. W« a n . . . r fl j for (Q B/ * a R— a RQ_ fl/ ), 

 like (Q fl/ R— RQ_ a ) above, must vanish, and by a like neces- 

 sity. In the product Q„ Q a/ , every function m n of Q flii affected 

 by the imaginary m 0) is of the form me, + tew, + M. In the 

 product Q„,Q a > the same term takes the form mw l + , wm l —M. : 

 let this be m' u ; and let Q „ become Q"« //} after all the substi- 

 tutions, m' n for m u , &c. are made. Then, reasoning as before 

 upon the equation Q'„,Q„=rt Q"„„— tf G R, where a Q Q llt =Q' ay 

 we can obtain a,, expressed in lermsofwab . . . ru\ l a' ll b' ll . . .r' // . 



