and Homoid Products of Sums ofn Squares. 4>55 



be made with two of these and any twelfth imaginary n, or m. 



In the case of thirteen imaginaries, then, we have the product 



Q Q =Q -j-R„, 



where R 3 contains, besides the six b Q i Q B,- . . g Q i Q G„ and the 

 six b l B t ... g l Q G„ another six, b Q n Q B n . . . g n Q G n ; but 

 it contains not the pair h Q l H t and i Q l Q l ti because the terms 

 (Hi) are now part of the function n n , and the terms (I,) of m lfl 

 by virtue of our new triplets. It is hardly necessary to observe, 

 that the six duad equivalents of b Q n Q . . .g Q n are deduced from 

 equating m Q n Q to the values of a Q in the first system of seven, 

 as in equations (B.). The reasoning (C.) applies here as before. 

 If there are fifteen ( = 2*4 + 7) imaginaries ab . . . . nop, R 4 , 

 in the product 



Qa Q« / =Q a „+R 4 » 



will contain, supposing that we add the triplet aop to the pre- 

 ceding fourteen, the eighteen terms of R 3 , and the two addi- 

 tional sixes, b p Q B p . . .g p G p , and h Q p Q H p , i p Q I p , . . . 



In the product of two pluquaternions of seventeen, R 5 has 

 the thirty terms of R 6 , and {aqr being the sixteenth triplet) 

 the two sixes b Q r B r . . .g r Q G r , and h Q r H r ...n r Q N r , 

 besides the additional pair o Q r Q O r and p Q r Q P r . And in the 

 product 



of two pluquaternions of nineteen, the condition-function R 6 

 contains, besides the preceding seven sixes, the two additional 



sixes b t B t g t Q G h and h Q t Q H t n Q t Q N, ; but 



not the pair o Q r Q O r and p Q r Q P r , as the terms (O r ) are now 

 part of the function t u , and (P r ) of s u ; the twenty quantities 

 w H a u . . . s tl t n being formed by the twenty-one triplets, 



■\-abc +ahi +aop 



+ ade—bdf+cdg +akl —hkm + ikn +aqr—oqs+2>qt 



+ afg + beg + cef; + amn -f- hln + Urn ; + ast -f ort + prs; 



if we suppose that the three systems are identical in form. But 

 this is not in anywise necessary, for any one of them, e.g. the 

 third, may be any one of eight systems containing the three 

 triplets a o p oi a q r oi a s Q t oi all of one sign : that is, the 

 function a n , indicated by the imaginary common to all the 

 twenty-one triplets, may have any one of the eight values, 



a n m aiv t + tea, ± (be, - cb,) ± {de, - ed^ ± (fg, -gf, ) 



+ ihif—ih,} + {k^—lkf} + {mn l —)im l } 



± [PPt-po^ ± [qr,-rq^ + Ut s -fej ; 



