and Homoid Products of Sums of n Squares. 457 



R' m be any function exhibiting a less number of imaginaries 

 than R m , the equation 



QaQ fl ,=Q a „+R'm 



is a contradiction, if ww, aa t .. . rr, are subjected to no limita- 

 tions. 



We shall next inquire how many possible forms there are 

 of the product of two pluquaternions Q a and Q U/ , or how many 

 different sets, w n a u b ip &c, functions of the 4« given positive 

 numbers imso l aa l . . . rr p Q fl// and the condition-function R can 

 be made, consistently with our definitions, to exhibit. Here 

 it may be useful to remark, that w lt is invariable, and that the 

 two first terms, which contain w and w n of any one of a tl b u ...r ni 

 are also invariable, through all the changes of form of the 

 pluquaternion Q a/ , so long as the values and signs of the given 

 real numbers are unaltered. 



We take first Q 0; , the product of two biquaternions Q tt and 

 Q a ,. Considering the set (be, de,fg) as one with {bc,fg, de) 

 and with (cb, ed, gf), but distinct from {cb, dc,fg) and from 

 (bd t ce i fg) 1 we can form sixty distinct sets of three duads with 

 the six imaginaries b o c o d e o f o g . Prefixing a to the three 

 duads of any set (c b d Q e f g )i the resulting triplets, a c b oi 

 «o d a toi a f o g 0i can be completed in eight different ways into 

 a complete system of seven. For example, the three just writ- 

 ten, considered as all of one sign, give for a n the two following 

 values, taking all the upper or all the lower signs together: 



a w =au>, + wa,± (<£,-&,) ± {de,-ed,) ± {fg,-gf) ; 

 to either of which values can be appended four distinct sets, 

 ^u c r A e ufuSi two of either type. 



Iherefore the biquaternion Q 0/; , the product of two given 

 ones, Q a and Q fl/ , has 480 distinct forms, each exhibiting a 

 different set of eight numbers, w u a„b u c a d„ e u f u g iP 



Let it now be supposed that two pluquaternions of nine, 

 Q a and Q n/ , are to be multiplied together. The product 

 Qa^ + Rj is determined completely, as soon as an eighth triplet 

 is added to a given complete system of seven. 



Thus, if we add to any one of the eight systems which give 

 either of the last written values of a lp the triplet a h o i o of the 

 same sign with a c Q b oi we obtain for a lt h n L the two sets of 

 values, 



a u = m t + wa, ± {cb,-bcd ± {de,-ed t ) ± (fg-gh,) ± {hi,- fh,) } 

 h H = hw t + , wh i + (iaj—aij), 



hi — iw i + xv> h ± { aJl i ~ h a i) 5 

 Phil. Mag. S. 3. Vol. 33. No. 224. Dec. 1848. 2 H 



