Extensions of Quaternions. 497 



[5.] As a second simplification of the general conception of 

 polynomes of the form (1), which will tend to render the laws 

 of their operations on each other still more analogous to those of 

 the quaternions, let it be now conceived that the choice of the 

 "constants of multiphcation," {fgh), is restricted by the follow- 

 ing condition, which may be called the " Law of Conjugation : " 



K.tt' = t't, orl^.ifig-=igif; .... (33) 



namely the condition that " opposite (or inverted) products of 

 any two of then symbols t,, . . t„, shall always be conjugate poly- 

 nomes." The indices / and g being still supposed to be each 

 > 0, the constants of multiplication {fgh), which had remained 

 arbitrary and disposable in [2.], after that first simplification 

 which consisted in supposing t,Q—l, come now to be still further 

 reduced in number, from (w + l)/i^to ^n{n~-^\). For we have 

 now, by operating with S on the equation (32), the following 

 formula of relation between those constants, 



(/^0) = (#)); (33) 



and by comparing coefficients of (/,, this other formula is obtained, 



-{fgh) = {gfh),iih>(i; . . . (34) 

 whence 



(//A) = 0,ifA>0 (35) 



Writing, for conciseness, 



im^ifg), {ff) = {f), . . . (36) 

 the squares, i^, of the n vector-units i, will thus reduce themselves 

 to so many constant scalars, 



t,^={l), ,„2=(2), . . ^/=(/), . . C' = («) ; . (37) 



and besides these, we shall have (/? + l)x— ^^ — '- = ^{ri^—n) 



other scalars, as constants of multiplication ; namely the consti- 

 tuents {fgh) of the polynomial expansions of all the binary pro- 

 ducts, ti' or LfLg, of unequal vector-units, taken in any one 

 selected ordur, for instance so that g > f; it being unnecessary 

 now, on account of the formulae of relation (33) (34), to attend 

 also to the opposite order of the two factors, if the object be 

 merely to determine the number of the independent constants, 

 which number is thus found to be ?i-[- l(/i^ — n)=i(n^ + n), as 

 above stated. Such then is the number of the constants of mul- 

 tiplication, including n of the form (/), and in(/t — 1) of the 

 form {fg), besides others of the form [fgh], which remain still 

 arbitrary, or disposable, after satisfying, first, the Unit-Laiu, 

 io= 1, and second, the Law of Conjugation, K . 1 1' = t' i. 



[6.] From this law of conjugation, (32), several general con- 

 sequences follow. For, first, we see from it that " the square of 



Phil. Mag. S. 4. No. 48. Suppl. Vol. 7. 2 L 



