Extensions of Quaternions. 495 



where the last term, or partial product, war', is now the only one 

 for which any peculiar rules are required. 



[3.] When the polynome P has thus been decomposed into 

 two parts, jij and sr, of which the one {p) is subject to all the 

 usual rules of algebraical calculation, but the other [■m) to pecu- 

 liar rules ; and when these parts are thus in such a sense hetero- 

 geneous, that an equation between two such polynomes resolves 

 itself immediately into tivo separate equations, one between parts 

 of the one kind, and the other between parts of the other kind; 

 so that 



if P = P', orj9 + w=J9' + cr', thenjo=/»', andCT = TO'; . (17) 

 we shall call the former part {p) the scalar part, or simply the 

 SCALAR, of the polynome P, and shall denote it, as such, by the 

 symbol S . P, or SP ; and we shall call the latter part (cr) the vector 

 part, or simply the vector, of the same polynome, and shall 

 denote this other part by the symbol V . P, or VP : these names 

 (scalar and vector), and these characteristics (S and V), being 

 here adopted as an extension of the phraseology and notation of 

 the Calculus of Quaternions*, in which such scalars and vectors 

 receive useful geometrical interpretations. From the same cal- 

 culus we shall here borrow also the conception and the sign of 

 conjugation; and shall say that any two polynomes (such as 

 those represented by p + vi andjo — ot) are conjugate, if they 

 have equal scalars [p], but opposite vectors ( + cr) : and if either 

 of these two polynomes be denoted by P, then the symbol K . P, 

 or KP, shall be employed to represent the other ; K being thus 

 used (as in quaternions) as the characteristic of conjunction. 

 With these notations, and with the recent significations of ju and ■or, 



p = S(j9 + w), w=V(jo-|-ot), j&--57 = K(;^ + ct) ; . (18) 

 or, wi'iting P and P' for^ + w and/? — ot, 



P' = KP, if SP' = SP, and VP'=-VP; .... (19) 

 and generally, for any polynome P, of the kind here considered, 



P = SP + VP, KP = SP~VP (20) 



We may also propose to call the n symbols t, . . t„ by the general 

 name of VECTOR-tiNiTS, as the symbol Iq has been equated in 

 (10) to the scalar-unit, or to 1 ; and may call that equation 

 (10) the UNIT-LAW, or more fully, the law of the primary unit, 



[4.] Already, from these few definitions and notations, a 

 variety of symbolical consequences can be deduced, which have 

 indeed already occurred in the Calculus of Quaternions, but 

 which are here taken with enlarged significations, and without 

 reference to interpretation in geometry. For example, in the 

 general equations (20), we may abstract from the operand, that 

 * See Lectures, />fl'5«Vn. 



