235 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
and the 3rd gives 
2__2 
C5— 21~i 7“3* 
We have thus abundantly verified the accuracy of the calculation, and there results 
the relation 
B=5Gi+yGr3+2G5. 
Lastly, let ot=6, 
f—ax^-{-Qhx^y-\-\hcxY-\-20dxy-{-\hexY-\-Qhxy^-{-ly^ 
(P==o 5^«+6|3^V+ + 15sa^y + 6^^+^?/® 
VL—u^.f—hu^ .y^x + 1 OM 3 .y^x^ — 1 Qu,y^x ^ + 5 Oyx^ —u^.x\ 
I shall here confine myself to the determination of a single argument in each of 
the terms M?; u,.u^\ u,m^\ u,.u, \ this will be ample for the purpose of 
verification, as the equation to be assigned is of the form 
B = C5 .G1-l-C3.G3d-C5.G5. 
The arguments which I select as the most simple, will be those expressed by the 
symbols (0, 1) ; (0, 2) ; (0, 3) ; (0, 4) ; (0, 5) ; (0, 6) respectively, then we have 
T5 = (ax -f %) d-Xj/) + &c . - ( ^3/) (“^ + 
= ([0, 5]d-...K+([0, 6]d-...)^+(-)3/' 
Q 5 = (mi .y - MaX) {u^y - Uqx) +&c. 
= (mi . Mg d" • • • d“ • • + ( • ' 
Hence supplying the binomial reciprocals 
1 
1 5 0 5 t , 
G5=([0, 5]d-...)Mi.M5d-|([0, 6]d-...)ai.M6d-&c. 
Again, 
T 3 = (ax® d- — ) (^^* + +&c. — {dx^ d- Sex^y d- 3hxy^ + ly^) (ax®d- • • • ) 
^([0, 3]d-...>'+(3[0, 4]d-...)x®yd-(3[0, 5]d-..-K/ + ([0, 6]-f ...)x®/d-&c. 
Q 3 = (mi . 3/® d- •) + 3^4^® d" 3 a 57 x® MeX®) &c . 
= (ai.a3d-..0^^-(3*^-^4+..0y^+(3ai.a5d-..0y^^-(“-^ed-..0/^* + &c., 
and the reciprocal binomial multipliers will be 
1 . — . — • — • &c 
1 ’ 6 ’ 15 ’ 20 ’ 
Hence 
G3=[0, 3]Mi.M3d-|[0, 4]Mi.W4d-|[0, 5]Mi.W5d-^[0, 6 ]m,.W 6 &c. &c. 
