VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 52S 
And we ought to have B™cG,+eG3, to satisfy which equation we must manifestly 
have c—4 ; to find (e), compare the coefficients ofu^, this gives 
4[0, 3]+24[l, 2]=f [0, 3] + ^[l, 2]+e(3[l, 2]-[0, 3]); 
accordingly we ought to be able to satisfy the two equations 
36 72 , 
-j-e=4 y+3^=24, 
each of which accordingly we find is satisfied by the equality e=~. 
Substituting in the equation for B above written, we thus obtain 
B=4G,+yG„ 
which will be found to be identically true. 
Art (70.). We may now see our way to a more concise mode of obtaining the 
numerical coefficients [by which they may in fact be computed and verified with 
comparatively little labour], connecting the Bezoutiant with the co-bezoutiant forms of 
the constituent scale. It will not fail to have been remarked, that throughout the pre- 
ceding determinations I have presumed the truth of the formula which admits of an 
immediate verification, that for all values of m and a we have the identical equation 
/ I ^ \ ^ f W 1 
{m.m— 1) .. . (m — 0 ) + 1) 
~ 1 
where 
Lo=Co.x’" 
CO — I 
L, Cj.x -\-{m cojc^.x^ “ ^.y-\-{m — u) — c^.x^ ^ ^ 
L. = (m — cy) c„+ , .y-f- (m— ^y) 
m — W— 1 
• ft nrflyi W — 2 np- I ^ 
2 1-2 -y ■••-rCm-y 
Let us now proceed to determine by an abridged method the linear relations corre- 
sponding to the cases of m = 5, m=Q, and first for m—b. 
Let 
f=ax^-{-b})x^y-\-\Ocxy-\-\Qdxy^bexy^Ji-hy^ 
(p=ax^-{-b^x*y -\- 1 OrxY+lOhxy-\-bixy^yy 
CL=u^y—4u^ yx -[-6^3 .?/ V — 4 u^ .yx^ y. 
In forming G^, G3, G„ let us confine our attention to the terms wf ; u,.u^. 
3 z 2 
