440 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO (px, fl 
where ai'e subject to satisfying the two equations 
^''.K2H-g’2 1 K 2 + .. .+^/fn-lK2 = 05 
so as to leave {n — 2 ) constants arbitrary ; and we thus obtain 
\- 3 ~ (§1 • K34-?2 MK-sd- ® + 
and so on, the number of independent arbitrary constants in ^ decreasing (as it ou^t) 
each time by one unit as the degree of S descends, until finally, if ro./-^o.?+^o-0, 
^0 being a constant, the general value for ^0 is found by making 
ro= (^ 1 ) . Qo+ (e2)Qi + • • • + (f«) • Q«-i 
^ 0 = — (g’l)Po“ (^ 2 )?! — • • •“ (f«) -Pn-lJ 
where ^ 2 , are subject to satisfy the (w- 1 ) equations 
{f,).K,+&c.=0 
(f,).K._.+&c.= 0 , 
which gives — 
Now evidently the lowest weight in respect to the roots of U and V that can be 
given to „_,K,K-'+&c., when the multipliers ^ 2 , .••?» are 
absolutely independent, is found by taking ^,= 1 ^ 2=0 ? 3 = 0 ...fn- 0 , which makes the 
weight of the leading coefficient in ^„_i, the same as that of K^, i. e. 1 . 
Again, when one equation, 
Ki+g’ 2 1 K- 1 + «-iK.i= 0 , 
exists between the (f)’s, the lowest weight will be found by making 
^;=,K, f;=o ^i=o...e„=o, 
which makes the weight of the leading coefficient in ^„_2 depend on 
K 2 -K, ,K 2 , 
which is of the weight 1 + 3, i. e. 4 in respect of the roots of/ and 9 . 
Similarly, will have its lowest weight when its leading coefficient is the determinant 
K, K 2 K 3 
.K 3 
2 K .1 2^2 2l^3J 
the weight of which is l+3+5=9; and finally, the lowest weighted value of + is 
the determinant represented by the complete Bezoutian square ; the weight m 
general of being 1 + 3 + ... + ( 2 /- 1 ), L e. P, or which is the same thing otherwise 
expressed, the weight of the leading coefficient of the lowest-weighted conjunctive otj 
and 9 of the degree I in x is (n-/)(m-;)^. It will of course have been seen in the fore- 
* n and m are supposed equal and t=7i—i. 
