and Homoid Products of Sums ofn Squares, 453 



we have 



4_ i e f— H] Cj 



± ek=fi=g/i; 

 and 



+ ceg=chk, 



+ ek=gh; 

 or 



ek=-ek; Q.E.A. 



It is therefore impossible that b i or c h Q should be equal 

 to k oi or to any other ninth imaginary; and the same contra- 

 diction is involved in the supposition that c Q i oi d Q i oi e i&j ' t Q 

 or ff z , or their equivalent duads in (B.), ftT£ any one of them 

 equal to a single imaginary. Each of these must therefore of 

 necessity appear, by itself or by its equivalent duad, in the pro- 

 duct of the pluquaternions Q« and Q a ,. Let these, e.g. be 

 pluquaternions of nine imaginaries ; then their product must 

 be of the form 



(w + a o a + b o b + e e+f f+g g + h h + i o i) 



X (iv, + a Q a ( + b Q b, + c c, + d d,+ e e, +f f t +g g, + K K + »« «/) 



=w u + a a u + bjb u + c c /t + d d tl + e e u +f f„ +g g„ + M//+ V« 



+ b i Q B. + c i C t +d i D t + * i E l +/^i F.+g i c G,; 



i. e. Q a Q„ ==Q a/ -f-Rj, or the product is a pluqualernion of 

 (2*1 -f 7) imaginaries, and an additional function of imaginaries 

 of a different form, which, for distinction's sake, we may call 

 the condition-function R 15 the number of its terms being the 

 number of conditions to be satisfied among the real quantities 

 of Q a and Q U/ , in order that the product Q„ Q U/ may be a 

 pluquaternion. In the case of nine imaginaries, there are six 

 of these conditions, viz. Rj=0, imposed upon the sixteen 

 numbers bb, cc,. . . is.. The terms h u and L of the pluquater- 

 nion Q fl// , are formed with the aid of the triplet a Q h Q i Q \ and 

 B 4 ., C f , &c. are given by the equations (B.), thus : 



B f = bi t — ib t + ch, — hc p C t = ci, — ic, + hb, — bh,, 



h lt m hw l -f ivh t + (ia, — ai,) , i a — m t + isoi, — (ah, — ha^j . 



Since + a b = c 0) and + a h Q = i o9 c h o =— i b gives 



a o b o h o = —a h b ,-* 

 or 



Mo=-Mo5 f . . . . (C.) 

 whence 



and the like property can be shown to belong to all the duads 



