420 
MR. SYLVESTER ON THE RESIDUES OF ^ EXPANDED 
and therefore M is of zero dimensions, and is the complete allotrious factor. la 
like manner if the complete residue corresponding to be 
a: 
or say M.L, the dimensions of L will be 
~ (e-}“ 1) — .4-j- (e-1-2 . 1)^, i.€. 1, 
and hence, as in the preceding case, M is of zero dimensions, and is the com- 
plete allotrious factor. 
Art. (5.). I proceed to show how the simplified residues may be most conveniently 
obtained by a direct process, identical with that which comes into opeiation in 
applying to the two given functions of x the method familiarly known under the 
name of Bezout’s abridged method of elimination. Let us call the two given func- 
tions U and V, and commence with the case where U and V are of equal dimensions 
(n) in X. The simplified /th residue will then be a function of ?i~i dimensions in .r. 
and of < dimensions in respect of each given set of coefficients, and may be taken 
equal to V,.U+U,.V, where V, and V, are each of (<- 1) dimensions in x. 
Let 
\] = af^.x”'-\-a^.x^~'^-\-a^.x'' 
V = + + -VK, 
we may write in general {m being taken any positive integer not exceeding /?}, 
U = aixr~ *+••..+ a^)x^-^-\- +....+««) 
Hence 

where if we use (r, s) to denote a^.b^ — a^.b,. for all values of r and s, we have 
„K, = (0,m-f 1) „K2=(0,m+2)-f (l,m-fl) „K3=(0,wz + 3) + (l,m+2) + (2,w + l), 
and in general ,„K,= 2(r, s), the values of r and a- admissible within the sign of 
summation being subject to the two conditions, one the equality r-\-s=m-\-}, the 
other the inequality r less than i. By giving to m all the difterent values from 0 
to m~\ in succession, and calling boX’^-{-b,x”'~''+ 
respectively Q™ and P„„ we have 
Q. .U-P, .V=K. 
Ivr" “+... 
K,. q 
Q, .U-P, .V=,K, 
..r"“*-l- 
iL.„ 
Q, .U-P, .V=,K, 
...+ 
2^.,, ^ 
Q._..U-P.-,.V=.^,K. 
-Jv, J 
