and Homoid Products of Sums of n Squares. 459 



three, a b c OJ a d e oi a f g ; and with a Q and the other six, 

 form also such a system, containing e. g. a k o i 0i a k l 0) a m n . 

 Then a n will be one of the eight, 



a u = aw, + wa, ±{bc- cb,) ±(de,— ed,) ± {fg, —gf,) 



± {hif — ik,} ± {klj — lk,} + {mn^nm,} ± [pp t —po^\, 



where the like brackets have all the same sign. If (Q a „ + R 4 ) 

 should reduce itself to a biquaternion, by the vanishing of all 

 the real quantities in Q a and Q fl/ except the twice eight 

 {wwpafib, cc l dd^ee l ff i gg^ i we should have before us in the pro- 

 duct only the eight functions (^ii a u^u c ii^ii e nfii§^ belonging to 

 that one of the 4S0 forms of such biquaternion, which is deter- 

 mined bythesystem of seven triplets madev/'\tha b c o d o e f o g ; 

 and if all the real quantities were to vanish from Q a and Q a , 

 except the twice eight (ww^afih/ifikp/mm^n^ we should have 

 before us one of the 480 forms of another biquaternion, deter- 

 mined by the triplets made with a h i Q k 1 m n . And, since 

 any one of the latter 480 may be combined with any one of 

 the former 480, and since in every case the resulting value of 

 a u may receive either of the increments + [pp—po^], according 

 as the triplet a o p Q is supposed of one sign or the other ; it is 

 plain, that we can form our product (Qa^+RJj so that o p o 

 shall be the duad excluded from both the systems of seven, 

 and that the two sixes shall be any pair of sixes, from which 

 a o and p are excluded, and that a shall be the imaginary 

 common to all the fifteen triplets, in 2 # (480) 2 distinct ways; 

 although, be it remarked, the resulting values of a u above- 

 mentioned will not amount to a number so great as this. But 

 any one of the thirteen, from which o and p are excluded, 

 may be the imaginary common to all the triplets, and deter- 

 mine, in a different way, to which of a H b n c lt) &c. the increment 

 + [pPf—poi] shall be added ; and we may combine anyone of 

 15*7 excluded pairs (o p ) with any one of 11*42 pairs of 

 sixes which exclude (o p ). Consequently, the total number 

 of ways in which the forty-six functions w ;/ « w ^ • . . . p tl may be 

 formed, in the product of two given pluquaternions of fifteen 

 imaginaries, is 2-(480) 2 -13-15-7'll , 42, or 290,580,480,000. 

 In all these cases, the condition-function R m is easily formed, 

 being given by the comparison of the fifteen equal values of 

 that imaginary which is common to all the fifteen triplets; 

 and the values o u and p lt , indicated by the pair of imaginaries 

 excluded from the two complete systems of seven, are given 

 by the single triplet, which is added to those systems. 



[To be continued.] 



2H2 



