Extensions of Quaternions. 133 



laws of the vector-units ikX are all included in this formula of 

 multiplication for any two vectors, such as 



p=.ix-{-Ky-\-\Zy p' = LCc'-{-Ky' + 7^ : . . (91) 



(A) . . pp'=z [m-^ — Iji^oca^ + (/jmi — mg^g) {y:^ + zy^) 



+ (m/ — 4/1)^2/' + {l^m^—m^mi) {za/ + xs^) 

 4- (^3^ — ^1 /a) -2'2^' + (/3W3 — ^1^2) (^y + yx') 

 + (i/i + Km^ + Xm^ {y^ — zy) 

 + (/c/g + Xmi + im^ [zx^ — xz') 

 -{-{\I^-\-tm^-j-Km{){xy' — yx')', .... (92) 



and it is clear that Quaternions^ are simply that particular case 

 of such QuADRiNOMEs (A), for which the six arbitrary constants 

 /j . . w?3 and the three vector-units c k\ receive the following 

 values : 



1^ = 1^ = 1^=1^ mi = m2=»?3=0, L = ij K=j, \=Jc. . (93) 



For the other associative quadrinomial system (B), which we 

 may call for distinction Tetrads, if we retain the expressions 

 (90) (91), we must replace the formula of vector-multiplication 

 {92) by one of the following form : 



(B) . . pp'= [Ix + my-\- nz) [W + m^i/ + nz^) 



+ {/cn—\m) (yz'—zy') + {\l—m){zx'—xz') + {cm—Kl) (xy'—yx') ; (94) 



involving thus only three arbitrary constants, Imn, besides the 

 three vector-units, o /c\; and apparently having no connexion 

 with the quaternions, beyond the circumstance that one common 

 analysis [12.] conducts to both the quadrinomes (A), and the 

 tetrads (B). 



As regards certain modular properties of these two quadri- 

 nomial systems, we shall shortly derive them as consequences of 

 the general theory of polynomes of the form P, founded on the 

 principles of the foregoing articles. 



[14.] In general, the formula (59) gives, by [2.], the two 

 following equations, which may in their turn replace it, and are, 

 like it, derived from the comparison of the vector parts of the 

 general associative formula, or from the supposition that A; > 

 in (54) : 



(fy)=X{ffeh){fhe),iie%ff; .... (95) 



=l^{geh){fhk),ifk%e,k%ff; , . (96) 



the summation extending in each from ^=1 to h=:n. Inter- 

 changing/and g in (95), we have 



{gf)=^{feh)(ghe),iie%f; .... (97) 



* See the author's Lectures, or the Philosophical Magazine for July, 1844, 

 in which the first printed account of the quaternions was given. 



