and Homoid Products of Sums ofw Squares. 501 



Q/»Q/'Q/Q,=<a i + R w+ S w + T m , 

 ■Q(«-s>Q(«-»)....Q'.Q.=<8t+R +S +T +...+Y, 



where R m S m &c. are in every line the same, as to the form and 

 number of their imaginaries ; inasmuch as each of them, when 

 first it appears, is supposed to contain all those of its own 

 degree, or equivalents of all those, which cannot be reduced 

 to simpler forms. This follows of necessity, although we 

 know nothing as yet about the number of terms in S m , T TO &c, 

 from the property already proved, that no duad b n , or r o v , 

 made with any two imaginaries taken from any two complete 

 systems of seven triplets, can be reduced to a monad, ox to an 

 imaginary of the first degree. In the product, therefore, of 

 three or of four pluquaternions, if b o n p and b i o r o v o be a 

 triplet and a quadruplet, of which neither contains two monads 

 taken from the same system of seven triplets, b n p oi or both 

 b n o p and b i r v , or else their equivalents of the same 

 degree, must be exhibited ; since no substitution can be made 

 that will reduce either of these to a multiplet of a lower degree. 

 And since /( = 6n + 1, or 6n — 3) imaginaries must be arranged 

 either in n complete systems of seven, or in (n— 1), and one 

 additional triplet, the product of n pluquaternions, Q, &c, 

 will exhibit Z m , a function of multiplets of the wth degree. 



If we now multiply the above expression for the product by 

 Q(% any (n+l)th pluquaternion of the same order, it is not 

 at first sight impossible, or improbable, that, since we have 

 only n complete systems of seven triplets at the most, all the 

 (n-f-l)plets of this product may reduce themselves to multi- 

 plets of a lower degree ; so that the same expression shall re- 

 present the product of (w+1), and consequently of (« + /*), 

 pluquaternions of the order i. 



The importance of the question, how far such reduction can 

 be made, will appear from the following reasoning. 



It has been already proved, that 



Q'.Q-^+R^ 

 Q_ t Q'_-a_-R w , 



and that 



by the mutual destruction of the imaginaries, independently 



