VIEWED WITH RELATION TO THE METHOD OP INVARIANTS. 
531 
T^i — {ax*-\-Ahx^y-\-QcxY-\-Adxf-\-eyA){^x*-\-Ayx^y-{.QlxY-\-Azxy^+ny^) 
— {ax^-\- A^x^y + 67 ^?"/ + s/ ) {hx* + + 6c?xy + Aexy'^+hy*) 
= [0, l]^®+4[0, 2]x^y-\-{Q{Q, 3] +)a;y+(4[0, 4]+)^y+([0, 5] 
+ 15 [I, 4] +20 [2, 3]yy+ &c. 
(where it may be observed that the numbers 15 and 20 in the coefficient of x\y* 
arise from the quantities 4^— -1 ; 6^— -4^). 
Again, Q,~VL‘^=u\.x^-\rQu,.u^x^y + \2u,.u,xy—Su,.u^xY-\-2u,,u,.xY^ See. 
Hence supplying the multipliers 
1 . Hi . 1 :z2 
’ 8 ’ 28 ’ 56 ’ ”^ 70 ’ 
we have 
18 4 
G,= [ 0 , 1]m?+4[0, 2 ]M,.W 2 +y [ 0 , 3]mi.M3+^[0, 
+^([0, 5] + 15[l, 4]+20[2,3 ])m,.M5. 
Again, the Bezoutiant 
B = 5[0, 1]«^?+2.10[0, 2]mi.M2+2.10[0, 3]mi.M3+2.5[0, 4]m,.m,+2.[0, 5]m,.M5+ &c. 
Accordingly, if we write B=Ci.Gi+C 3 .G 3 +C 5 .G 5 , we have, as above remarked, c ,=5 ; 
and to determine C 3 , c^, we have, by comparing the coefficients of Uj.u^, u^.u^ in 
B, G„ G 3 , Gs, 
20=y + C3 
10 = y + C3 
These two equations, then, as it turns out, are not independent, but are satisfied 
simultaneously by 
50 
Finally, equating the coefficients of the several arguments in u^.u^, we have 
^35"^y from the argument [O, 5] 
15 50 1 
argument [l, 4] 
argument [2, 3]. 
The 1 st of which equations gives 
the 2 nd gives 
1 ^_14_2 
I A_£ 
Cs — 7 + 21— 3 J 
