Extensions of Quaternions. 131 



And hence (writing k instead of J), we see that the trinome^, 



T = z + ix-\-Ky, ...... {76) 



where xyz are ordinary variables, will possess all the properties 

 of those polynomial expressions which have been hitherto con- 

 sidered in this paper, and especially the associative property, if 

 we establish the formula of multiplication, 

 {tx + icy) {ix' + Ky') = {bx — ay) {bx' — ay') 



-{■{aL-{-bK){xyf-yx'); .... (77) 



wherein a and b are any two constants of the ordinary and alge- 

 braical kind. In this trinomial system, 



s'l + 1^1 _j_ Ky" =(2 + Lx + fcy) [z' + 1<2/ + Ky'), . (78) 

 if 



x" =zx' + z'x + a {xy' —yx'), -^ 



y" = zy^+z'y + b{xy'-yx'), I . . . . (79) 

 z" = zz' + {bx '— ay) {bx' — ay') ; J 

 we have therefore the two modular relations, 



z" + bx"-ay"={z-{-bx-ay){z'-{-bx'-ay'), \ 

 zti _ ba;'f + ay" = {z-bx + ay) {z' - bx' + ay') ; J ' ^ ^ 

 that is to say, the functions z+ {bx — ay) are tuw linear moduli 

 of the system. A general theory with which this result is con- 

 nected will be mentioned a little further on. Geometrical inter- 

 pretations (of no great interest) might easily be proposed, but 

 they would not suit the plan of this communication. 

 [12.] For the case n = S, or for the quadrinome 



r =XQ-f- L^X^-\- 12^2' ^3'^3) .... \y^) 



we may assume 



8*2%= ^1, 8*3*1 = 62, 8*1*2=^3, J 



and 



7*2*3= — V%*2= hh + *2^3 + *3^2J ^ 



763^1 = — 7*1*3= *2/2 + *3?Wi 4- *iW3, i* . . . . (B3) 



V*i*2 = — • V*2*i = *3/3 + *iW22 + ^2^1 '> J 



and then the ^{n'^—n^)=27 scalar equations of condition, in- 

 cluded in the vector form, 



V(*.*VO=V(**'.*"), (84) 



are found on trial to reduce f themselves to 24 ; which, after elimi- 



* I am not aware that this trinomial expression (76), with the formula 

 of multiplication {77), coincides with any of the triplet-forms of Professor 

 De Morgan, or of Messrs. John and Charles Graves : but it is given here 

 merely by way of illustration. 



t The reason of this reduction is exhibited by the general analysis in [14.]. 



K2 



