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422 MR. SYLVESTER ON THE RESIDUES OF ^ EXPANDED 
Let U=a..3;**'+a..T*“-’+a..A'*'"'+ &c. 
Making , , , , . 
U = + + +&C. +Cr„+e) 
we obtain the equation 
Q..U-P„„.V=.K,..t'+-'+»K,.jf"-’+&c-+..K„ (I2-) 
where 
Q^=(&,X“+ . . . +5JPe+«”K-^*'"’”+ • • • +«P+J 
Bv giving to m evei-y integer value from 0 to (»-l) inclusive, we thus obtain u 
equations of the form of (12.), each of the degree n+e - 1 in x, and of one dimension 
in regard to each set of coefficients. 
In addition to these equations we have the (e) equations of the form 
-{■hy', ( 13 -) 
in which ^ may be made to assume every value from 0 to (e- l) inclusive and the 
left right-hand side of the equation for all such values of will remain of a depee 
in T not exceeding n-f e- 1, the degree of the equations of the system ahove described 
There will thus be (e) equations in which only the (b) set of coefficients ap|.ear, and 
(«) equations containing in every terra one coefficient out of each of the two sets 
The total number of equations is of course n-l-e. Between the (e) equations of t.ie 
second system (13.) and the (r) occurring first in order of the first system we 
may eliminate dialytically the e+r-1 highest powers of x, and there will thus ause 
an equation of the form 
4^_,U-s.,+,-,.V=La:-’-+L'ir”-’-'+ &c. -fL (14.), 
where 6.., and are respectively of the degrees r— I and e-fr-l in .r, and 
L L' (L) are of (r) dimensions in the (a) set, and of (e+r) dimensions in tie w) 
oi'coefficients, aU consequently Lt.- + L'.--+... + (L) ^ 
ditions necessary and sufficient to prove its being (to a numerical factoi ;-.es) a 
simplified residue to (U, V). 
Thus suppose 
V= by- yb^x +b^. 
Then, corresponding to the system of which equation (13.) is the type, we have 
y = bo.x^+b,.x-\-b, 
x\=bo.x^+b,.x^yb,.x. 
