1905 * 6 .] Dr Muir on the Theory of Alternants. 
359 
which Lame had just published was originally given by Binet in 
1811. He thus has occasion to say that the form under which he 
had considered the equation of homofocal surfaces was 
a 2 . b 2 r 
B 4 " 
K-A + K- B ' K.-C 
where a, b, c are the co-ordinates of any point on the surface, 
A, B, C are positive constants such that A>B>C, and K is a 
quantity which may be of, any magnitude greater than C. And as 
Lame had obtained expressions for the co-ordinates in terms of 
three values given to K, Binet intimates that many years before he 
had not only done the same but had extended the solution to the 
case of n equations. It is this purely algebraical problem which 
is of interest to us, and fortunately Binet gives it in full. 
Taking the set of equations in the form 
a 
K-A 
= 1 3 
+ K-b + K-C + ; ■ 
. . =1, 
+ K 1 -B + K 1 -C + * ' 
. . =1 , 
b c 
+ k 2 -b + k 2 -g + ' ' 
. . =1, 
he introduces, for temporary purposes, two functions, E(flj), f{x\ 
the former being 
(x - A)(x- B)(# — C) ..... 
and therefore of the w th degree, and the latter being any integral 
function of a degree less than n. He then recalls the fact that 
f(x) -r F(#) can be partitioned into n fractions having x — A , x - B , 
x — C , . . . for denominators, the result as given by Euler being 
m _ /(A) . /(B) /(C) 
1» - (x - A)/ (A) {x - By'(B) (x - G)f(C) 
Substituting successively K, , K 2 , . . . for x' in this, a set of 
equations is obtained from which it is seen that the solution 
of the set 
-^+^+^+ 
K-A K-B K-C F(K) ’ 
a ■+.,» -| ,r .|. 
K, - A Kj-B K, -C ‘ • F(Kj) ’ 
