528 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
Hence supplying the binomial reciprocals 
1 ; 2 ’ 
we have 
3]-3[1, ■2])(u,.u.-uI) + 2{[0, 4]-2[1, 3] 
+ ([1, 4]— 3[2, 
Again, 
T,=(ax>+3l>^^+3cief+df)(l3x‘+3yJ^i/+Sixf+if) 
_(a:<;»+ 3 ( 3 x>+ 3 yJ 3 /’+y)(l'u^’+ 3 c^'t/+ 3 <?*y+«/’) 
= [0, l]^'+3[0, 2]a:‘r/+(3[0, 3]+6[l. 2])a;y+([0, 4]+8[l, SjK/ 
+ (3[1, 4]+6[2, 3])*'t/‘+3[2, 4]/, 
and Qi=n^ 
„ ^2 ,2/®— 6m, . u ^. x ^’ y -^- (9 m 1 + 6Mi .1/3)3/ V— (2mi .1I4+ 1 81/2 .1^3) 
+ (9 mI+ 6M2 .M 4)3/'«'“ 6M3 .M4 . yx ^-\- u \. x \ 
Hence, supplying the reciprocal binomial coefficients. 
we 
find 
G,= [0, l]»5+3[0, 2]t.,.«,+ (5[0, 3]+|[l, 2])(9a>.+6«,.M.) 
+ (i[0, 4]+^[l, 3 ])(m,.m,+9m,.m.)+(5[1, 4]+|[2, 3]) X (9mH-6u,.u‘) 
+3[2, 4]m,.M4+[3) 4]“4. 
Now the Bezoutic square, taking account of the binomial factors in/ and 9, may be 
written under the form 
[0,4] 
; 4[1, 4] 
4[0, 1]; 
6[0, 2] ; 
4[0, 3] ; 
6[0, 2] ; 
4[0,3] - 
r "I. 
L+24[1,2]_ 
■ [0,4] 1 
’ !_ + 16[l,3lJ’ 
r ti>4] 1 . 
4 [0,3] ; 
_ + 16[l,3]_ 
_ d-24[2, 3] _ 
[0, 4] ; 
4[1,4]; 
6[2, 4] ; 
[3, 4], 
Hence the Bezoutiant B becomes 
4[0, 1]m?+ 12[0, 2] Ml. 1/2+ (4 [0, 3]+24[l, 2])i/l+2[0, 4]i/ii/, 
+ (2[0, 4]+32[l, 3 ])i/2.i/3+8[1, 4]i/2.m,+ ([1, 4]+24[2, 3])i/l 
+ 12 [2, 4] 1/3. 1/4+ [3, 4] 1/^4. 
