530 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
A comparison of the coefficients of these with those in the Bezoutiant (B) trill be 
sufficient for assigning the three numerical quantities which connect B wi - 
I omit because G. is the only one of the G’s for any value of (m) which con- 
tains u\ or u,.u„ and in G, the terms containing u\ and u,.u, are 
[ 0 , + 2 ]. Ml. M 2 , 
and the corresponding part of the Bezoutiant is 
M7.[0, l]M?+7M.(m— 1)[0, 2 ]mi.M 2 ; 
so that if we write 
B=Ci.Gi+C3.G3+C5.G5+ &c., 
the two terms nl and will only enable us to form one equation with the c's, viz 
c =m Again, instead of considering the entire coefficients of u, .u, and «, it will 
be sufficient to take a single argument of either of these coefficients (in the forms to 
he compared), as for instance [0, 3] and [l, 3], Then c. being known, c., r, will be 
determined ; but for the purposes of verification I shall furthermore compute t e 
whole of the coefficient of 
Accordingly [calculating the G system in reverse order] we have 
G,= { [ 0 , 5] -5 [1, 4] + 10[2, 3] } .3ul) 
= { [ 0 , 5] — 5 [l, 4 ] + 10 [ 2 , 3 ] }Mi.M5+ ••• 
£3.9= &c. &c. ; 
— {3(bx^+2cj;i/+ dy"^) -\-2lxy-\-zy‘^)~{ (3x^ +2yxy+ ^y") (cx^-i-2dxy+ ey ^) } 
= [0, 3]^"+(2[0, 4] + ...Ki/+{[0, 5] + [B 4] -8 [2, 3]}(cy+&c. 
[The number --8 results from the calculation 1 — 3(4— 1) = — 8.] 
Again, 
E3a=(My-2M,3/x+M3.a(^)r-2(M,.y-2M3.3/x+My)|^+(M3y-2M,.3/fr+My-);j\ 
. Q 3 = (Mq .y^ — 2uyyx + u^x^) (Ms/ — 2uyyx-\-u^x^) — {u^ .y^ — 2 M 3 .yx + m^ . x-)\ 
= Ml . Ms - 2 u,.u,.y^x+u, .u,fx^ + &c., 
all the terms and parts of terms unexpressed being free of Mi, and therefore not 
necessary for our purpose. Hence supplying the reciprocal factors 
1 ; -7 
1 1 
4 ’ 6 
we have 
G,= [0, 3]m,.«.+ ([0, 4] +|{[0, 5] + [I, 4] + [2, 3 ]}M,.«,+&c. 
Again, expressing E../and E..f in the usual way, we obtain 
