VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 
533 
Finally, 
T, = + &c.)(/3j7®+5yx^3/+ — &c. 
= ([0, l] + ...y<>+5([0, 2] + ...)a?93/+(10[0, 3] + ...)a^y+(10[0, 4]4-...)xy 
+ (5[0, 6] + ...)a?y4-([0, 6] + ...)xy+&c. 
+ (lOMi.M5+...)3/V4-(2Mi.M6+-..y‘*'®+ ; 
and supplying- the numerical series 
1. _1. 1 . J_. ^.o 
’ 10’ 45’ 120’ 210’ 252 ’ 
we have 
G,= [0, l]Mf+5[0, 2]u,u^-Jr~\0, 3]mjM3+|[0, 4] 
5 1 
+ ^[0, 5]m,.M54-— [0, 6]mi.M6+ &c. 
Again, the Bezoutiant 
=6[0, ]]m?+30[0, 2]m..M3+40[0, 3]m, .M3+30[0, 4]^,.^^ 
+ 12[0, 5]m,.M5+2[0, Q~\uy.UQ-{- See. &c. = B. 
Hence making 
B = C,.G2 + C3.G3+C5 .Gs, 
from Ui and Mj-Mj we obtain respectively 
Cl = 6 
5c, = 30 ; 
hence from Uj.Ua and u^.u^ we obtain respectively 
240 , . 
9 + C3=40 
30 , 3 
2 I 2^3 — 
40 
> or C 3 = y ; 
hence from Uj.u^ and u^.ug we obtain respectively 
l+"^=12, i.c. C3=12-8-^=^" 
hence 
, 40 11 „ . 1 ^219 
Xl26+ 3 •20"^2^5 — 2, ^.e. gCg — .. — — ; 
— 1^ 
Cs — 
and the equation sought for is 
B — 6 Gi + -3- Gg + y Gj. 
Art. (71-). The following table exhibits the relations between the Bezoutiant and 
