458 The Rev. T. P. Kirkman on Pluquaternions, 



and the function R,, the same for either set, is deduced from 

 the equality of the four values of a Q . If the eighth triplet is 

 a o *o^oj we obtain the two sets 

 a u - aw, + wa, ± {cb, - be,) ± (de, - ed]) ± (fg, -gf,) ± {ih, - /?/,), 

 h u = hiv, -f isoh t + (ai t — iaj), 

 i n = iw, + ivi, ±(ha,— ff^,) ; 

 and a different function, R 15 is given by the equal values of a Q , 

 and is the same for either set. 



To each of these four sets of a n h u i n can be appended four 

 distinct sets of b n c H d n e H f n g lp two from either of our two types. 



But it is not necessary that either a h i or a i Q h Q should 

 be the eighth triplet which is added to the given complete 

 system of seven. It may be either f h i or f i h oi for ex- 

 ample. Either of these will add to the value of/] y , which is 

 exhibited by the biquaternion Q„ /; (made by the given system 

 of seven), a binomial function of liih^i^ while no such addition 

 will appear now in a n ; and the values of h n and i lt will now 

 contain^ andy], but not a or a r The function R l will in 

 either of these two cases be formed by the comparison of the 

 four equal values of/ . 



It is clear then, that, if h Q i be the two imaginaries which are 

 excluded from the given complete system of seven, we can add 

 to any one of the seven terms (a n ...g /y ) of any one of the 480 

 biquaternions that can be made with the given complete system 

 of seven, either of the quantities ± {hi^ih^ and append to the 

 eight quantities w^a^ ..•£,/, thus increased, the corresponding 

 values of h u ik and R, ; that is, we can construct the sixteen 

 functions w ll a u b IP Si-c. in theproduct Q 0/I + R 15 in 7'2*480 ways. 

 If a Q b Q were the excluded duad, we should have another series 

 of as many different sets of w tl a tl b jP &c. ; and, in fact, the 

 total number of ways of forming the product of two pluquater- 

 nions of nine is 36 , 7 , 2'480 = 241,920. 



Since the case of fifteen or (2 4 — 1) imaginaries possesses some 

 interest, appearing, as it does, to follow next in order after 

 those of (2 2 — 1) and (2 3 — 1), it may not be considered mere 

 learned trifling, to assign the number of ways in which (Q ff// + R 4 ), 

 the productof two pluquaternions of fifteen, can Deconstructed. 

 This product is completely determined, when to two given 

 complete systems of seven triplets is added a fifteenth triplet, 

 containing an imaginary which is common to both the systems 

 of seven. Let a Q be that imaginary ; and of the fourteen, 

 b c . . . m n , take any six, b Q c d e o f g oi and any other six, 

 h i k l m w , o p being the duad excluded from both the 

 sixes. With a Q and the first six, form a complete system of 

 seven triplets, of which the three containing a shall be any 



