Extensions of Quaternions. 493 



of P', and cr"o . . a,"„ of P", are subject to all the usual rules of 

 algebra, and to no others j but that the other symbols, Iq . . *„, 

 by which those constituents of each polynome are here symboli- 

 cally multiplied, are not all subject to all those usual rules : and 

 that, on the contrary, these latter symbols are subject, as a 

 system, to some peculiar laws, of comparison and combination. 

 More especially, let us conceive, in the first place, that these 

 n + 1 symbols, of the form Lf, arc and must remain unconnected 

 with each other by any linear relation, with ordinary algebraical 

 coefficients ; whence it will follow that an equality between any 

 two polynomial expressions of the present class requires that all 

 their corresponding constituents should be separately equal, or that 



if P' = P, then x'q=Xq, x\=:x-^, . . x'„ = x„ : . . (2) 



and therefore, in particular, that the evanescence of any 07ie such 

 polynome P requires the vanishing of each constituent separately ; 

 so that 



if P = 0, then^o=0, a;, = 0, ..a^„ = 0. ... (3) 



In the second place, we shall suppose that all the usual rules 

 of addition and subtraction extend to these new polyuonies, and 

 to their terms ; and that the symbols i, like the symbols x, are 

 distributive in their operation ; whence it will follow that 



P' ± 1' = *o(^'o ± ^o) + • • + ^n (*''„ ±Xn), ' ■ (4) 



or that 



^ix'±'Ztx = 'Zi{x^±x): (5) 



and as a fiu'ther connexion with common algebra, we shall con- 

 ceive that each separate symbol of the form t may combine com- 

 mutatively as a factor with each of the form x, and with every 

 other algebraic quantity, so that tx=xi, and that therefore the 

 polynome P may be thus written, 



V=Xq1.q + Xi1.i+ .. + x„i,„ = 'S,xi.. ... (6) 



But, third, instead of supposing that the symbols t combine 

 thus in general commutatively , amomj themselves, as factors or as 

 operators, we shall distinrjuish generally between the two inverted 

 {or opposite) products, a' and I'l, or ifig and igi/i and shall con- 

 ceive that all the (m + I)^ binary products {u), including squares 

 {t^z=:u), of the n + 1 symbols l, are defined as being each equal 

 to a certain given or originally assumed polynome, of the general 

 form (1), by (/i + l)^ equations of the following type, 



v*^=(/^o)*o+(/^i)^+ • • +(/z//')^A+ ■ ■ +(/^«)*»; • (7) 



the (n + 1)^ coefficients, or constituents, of the form (/j/Zi), which 

 we shall call the " constants of nuiltiplication,^' being so many 

 given, or assumed, algebraic constants, of which some may vanish, 

 and which we do not here suppose to satisfy generally the rela- 



