VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 627 
Supplying for facility of computation the reciprocals of the binomial coefficients to 
the index 4, viz. — 
1 - _1. 1 . 1 
^ ’ 4 ’ 6 ’ 4 ’ 
we obtain 
G,= (a^-6cc)u^,-^2{ay-coc)u,.u,-i-(2(by~c[^)-\-^(ah~dcc))ul 
It will here and henceforth be more useful to employ [r, to denote, not the 
1 erence of the cross products of the (r 4 -l)th and (^+l)th entire coefficients in 
/and <p, but the difference of the cross products of these coefficients divided each by 
Its appropriate binomial coefficient. We may then write 
G. = [0, l]ul+2l0, 2]u,.u, + ([i, 2]+i[0, 3])u,.u,+ (2[l, 2]+|[0, 3]).ul 
+ 2 [l, 3 ]m2.W3+[2, 3]m|. 
Again, 
G,= {(aS-&)-3(iy-c(3)} + («..tt 3 _„|) = (|;o. 3 ]- 3 [ 1 , 2 ])(«.m,)-([ 0 , 3 ]- 3 [ 1 , 2 ]K. 
Hence 
G,- 3 G,= [0, 1]«;+2[0,2]m •!'.+2[l,2]!<,.M3+([0,3] + [l,2])«|+2[l,3]a3.M3+[2,3] 
But, again, the Bezoutiant ofy, tp corresponds to the matrix 
3[0> 1] ; 3[0, 2] ; [0, 3] 
3[0, 2 ]; [0,3]+9[l, 2 ]; 3[1, 3 ] 
[0, 3] ; 3[1, 3] ; [ 3 , 4 ]. 
Hence summing the sinister bands to form the coefficients, we have 
B=3[0, l]«;+6[0, 2 ]m;,M 3+(3[0,3]+9[I, 2])«»+6[1, 3]a3.«,+ [ 2 . 3]«‘=3G,-G.. 
3rd. Suppose m=4, 
/=ax*+4bxy-j-6cxy+4dxf-i-e^* 
p=ciX^+4^x^^-^6yxY-i- 
rp, G,=^uy—3uyx-\-3uYyx^ — u^.x^. 
^^•f={ax-\-hy)l^^^hx-\-cy)l\-\-3{cx-^dy)ln^-\-{dx-\-ey)n\ 
\~{ax-{-^y)[dx-\-ey)] \~{^x-{-Yy)(cx-\-dy)\ 
=([0, 3J-3[], 2j>^+([0, 4]-2[l, 3-\)xy-\.{\\, 4]-3[2, 3])/ 
Qs = (u, .y—u,x) Yy-u.x) - (u,y—u,xy 
= {U\.u^—u'^y'^{u^.u^--u^u^xy-\-(u^,u^--u\)x'^. 
MDCCCLIII. q „ 
