376 
Proceedings of Royal Society of Edinburgh. 
in every other case. The determination, we are told, can he 
made by using an extension of Lagrange’s interpolation-formula, 
the outcome of the work being 
y\ _ m / _ i un(n+i) H(^ , , . • • , C) ■ U(a , , a 2 , . . . , CL n ) 
f\t) -Ah) -/(y aq 
which, of course, gives us 
D = T-A. 
This relation having been established, Borchardt then proceeds 
in a line or two to use it for the main purpose of his paper. As 
the determinant D, he says, arises out of the determinant by 
performance of successive differentiation with respect to all the 
variables f , t 1 , t 2 , . . . t n , there is obtained at once an alternative 
expression for T , namely, 
T _ / _ 1 y.+i /(0 -/(<i) • • ■/(*«) 0 3 d_ ( n(e , tj , ■ ■ ■ , f„) \ 
v n(< , <j t n ) • • a<„ • • •/(*„)/ 
or say rather 
nc, t a , . . ., t n ) 
'T' _ / _ I V/i+1 bt 0^ dt n f(t) • /(tf . . . f{t n ) . 
1 ' n(Mi, 
/(*) •/&) • • ■/(*.) 
and so the transformation aimed at is accomplished. 
Prouhet (1856, March). 
[Note sur quelques identites. Nouo. Annates de Math., xv. pp. 
86-91.] 
In order to generalise certain algebraical identities published 
by 0. Werner in Grunert’s Archiv , xxii. p. 353, Prouhet first 
establishes the theorem in alternants foreshadowed by Prony and 
Cauchy, and readily derivable from Schweins’ first multiplication- 
theorem. His mode of treatment may be concisely stated as 
follows : — 
To say that a , b , c , d , e , / are the roots of 
x 6 - ppc b + p 2 x* - p z x? +pjX 2 - pyx + p 6 = 0 
implies that 
p 1 = 2a , p 2 = lab , p s = 2a5c , . . . ; 
