( 447 ) 
BOECHAEDT'S FOEM OF THE ELIMINANT OF TWO 
EQUATIONS OF THE nth DEGEEE. 
By Thomas Mum, LL.D., F.E.S. 
(Eead September 15, 1909.) 
1. The problem which Borohardt set for himself was to express the 
eliminant of the equations (p{x) = 0, \l^{x) = 0, both of the nth degree, in 
terms of 7i -\- 1 arbitrary values of x and the corresponding values of 
0(£c), \p(x). To this end he made use of the result of Cayley's mode 
of finding Bezout's condensed eliminant ; and as a consequence his 
solution took the form of an expression in terms of the ^n(ii + 1) different 
values of 
y-x 
found by giving x and y the said n -\- 1 arbitrary values, say, oo, a^, «„. 
He experienced difficulty, of course, when x and y had to be given one 
and the same value ; but this was overcome by establishing the theorem 
that If (()(x), xpi^x) be rational integral functions of the nth degree, and 
\(f){x)\p{y) — (p{y)\l/(x)^^{y — x) be denoted by F(ic,2/) and {x-ao)(x — aj) 
...{x — a„) by f(x) , then 
F(a,. , a,.) _ _ F(a^, a,) 
f'i»r) ~ ZjsTwy 
where s is given in succession all the values 0, 1, except r. In other 
words, he succeeded in showing that each of the n -\- 1 illusory forms 
becomes known as a consequence of knowing n of the other forms/'' 
2. The mode of proof given by him is laborious, the two given functions 
being expressed in the interpolational form 
r=n r=n 
y/Xxyix-a^ 0(„,), y/M/i^ +(„,), 
Zj /'(«.) Zj I'M 
7-— 0 r= (J 
* For marked advances recently made on Borchardt's work the reader is referred to 
two papers by Mr. A. L. Dixon in the Proceed. Lond. Math. Soc. (2), vi. pp. 468-478, 
vii. pp. 49-69. 
