454 MR. SYLVESTER ON THE 
RESIDUES AND SYZYGETIC MULTIPLIERS TO <pJ, /j 
which is the same thing, to ^*. Conversely, it is obvious that if v-e adopt as our 
primitive functions c/(m) and (c) being the value of K when / = 0 we shall 
obtain as the general form of our syzygetic equation, bearing in mind that {m 1) 
now replaces (n), , 
and similarly, if we adopt as our primitive functions and c%, we obtain lor our 
general syzygetic equation, observing that (n - I) now replaces (m), 
so that (making abstraction cf the constant factors and looking merely to the torms 
of the several functions which enter into the equations) we see that on the hr.st 
hypothesis, viz. of being substituted for 9 . the conjunctives of each degree in .1 
change places with the second conjunctive factors, 2 . e. the original multiplieis ot c 
of the same degree in and vice versd-, and in the second hypothesis, where r 
takes the place of fm, the conjunctives of each degree in a; change places wit t e 
first conjunctive factors, k e. the original multipliers of/ of the same degree in .r, and 
vice versd-, and being respectively multipliers of J and / such t lat ti.c 
difference of the respective products is independent of x. These lesu ts oug o e 
capable of being verified by aid of our general formulae for t, r, and as this verm- 
cation will serve to exhibit in a clearer light the nature of the reciprocity between 
the conjunctives and the conjunctive factors, it may be not uninteresting to set it 
"""l-t. (33.). As usual, let h, K-.-K be the roots of/(/, and the roots of 
®(.r), the last conjunctive factor to (p, which is of the degiee (?n— 1) in .r, vi 
represented, neglecting powers of ( — ), by where 
= 2 (^ - /i„) (a; -h,)...{x- /q,„_,) 
55, % 
■ •n. 
<ln 
If now we for greater simplicity make = and call the roots of /..n;',. 
any such quantity as 
Qm 
^ I ^2--’^ m-i . 
, , T V'q^/ - T \ q.2- ' ' 9ui-l' 
1 
= R 
Qm' 
. Since i ie .l«njs supposed less than « (« being the degtee of the lower degreed of the two functions/ and do 
the tact ot the last gnotient to being wanting to ^ 
the text above, since this latter will contain a, many quotients as can in any case be required for expressmg 3,. 
