20 REPORT— 1873. 



Now the coeHiei(mt of any temi x"~' in the lirst part of the above equation 



1.2..» 



(n-J)(n-2)..(n-i+l) . {n-\). .{n-i-^Y) .^ .^ 

 = n % n /■• IN = t'i 1 o ■• = « [«. 1 1 say. 



1.2.. (.--I) 



Hence (2) may be written thus, 



(da, a'i>!c+^b, 26^,r+se,. .) (z, 1)'' = 0; 



ta -&b 



or patting j-,=ff', -j-. — o, . ., (-3) becomes 



(a',a+b;2h-i-c',..)(.r,lf=0. . 

 Now the resultant of (1) and (4) with respect to .r is 



a [ii, l'\ b 



«' [«, I](6'+«) 

 a' 



■«, 21 c 



n, 1] & 



a 



«,2l(e'+2J) . 

 >, 1](5'+ «) . 

 u' 



-0. . 



(3) 



(4) 

 (•5) 



And if anj' one of the minors formed from the » upper lines of (.5) be represented 

 by F (r/, 6, ..), and the complemeutary one formed from the n lower ones by 

 F, {a', b'+a, '. .), and if fm-ther we write F, Fi for F {a, b, . .), Fj («', b', . .) re- 

 spectively, then (5) may be written thus : — 



0=2r («, b, ..)¥, («', b'+a, . .) = 2FF,+2vFF, + 2j^FF^+. . , . (6) 



The last two terms of (G) offer some peculiarit}'. In fact it is not difficult to see, 

 by reference to (5) and (7), that the last term, viz. S,--^ — -''FFj is =a^D, where 



□ is the discriminant of (1). Also if we multiply (G) throughout by ?a;", the last 

 term but one divided by the last will be the coefficient of ?x" in an equation 

 for determining d^ ; in other words, it will be = —2dx=n'd(-\=-^(a^b — b^a). 



1 n—i 



So that 1 2 (w-1 )^^ FF, = n(a-i>b-b<ia)a; 



and the last two terms of (G) are consequently 



^ a ' 



(8) 



Consider the cases of h=2, and m=:3. For the quadratic («, b, c) {x, l)- = 

 (6) takes the form 



4a2(af— 6-)+8(flc— 6-') {oV —db) — \(bd -h'c) (ab'—a'b) + {ac'—a'c) =0; 



and if we subject the variability of the coefficients to the single condition ub' — ab 

 = 0, the resultant reduces to 



n c/.v a — * ' -" 



whence 



^•'■=±5 



rt c 



— ,, ^ -; 



s/(lfi^i(c) a 



