298 
Proceedings of the Royal Society of Edinburgh. [Sess- 
in vol. x. of his journal, but such is not the case, the subject being not even 
referred to in that volume. 
Prouhet, E. (1852). 
[Memoire sur quelques formules generales d’analyse. Journ. ( de 
Liouville) de Math., (2) i. pp. 321-344.] 
The third and last section (§§ 40-45) of Prouhet’s memoir is headed 
“ Theoremes sur quelques determinants de fonctions.” The first theorem 
proved is that just mentioned as having been used by Malmsten. He then 
takes the set of m+1 equations, 
x 0 . d°(ucj> 0 ) + x 1 . d\u<$P) + . . . + x m . d m (u<h°) — d m+l (n<f>°) 
x 0 . d 0 (ucf> 1 ) + x 1 . d l (u<f> 1 ) + . . . + x m . d m (u<f> 1 ) = d m+l (u<ji l ) 
x 0 .d°(ucf> m ) + x l .d 1 ( 2 i<j> m )+ . . . + x m . d m (v<f> m ) = d m+1 (u<l> m ), 
and with the help of the said theorem obtains at once 
x m = d ~=d(logA), 
where A stands for 
| d^iii^.d 1 ^^) d m (ucj> m ) | . 
Next, by using in connection with the same sets of equations the multipliers 
± cf ) m , , \m{m 4- 1)<£ 2 , -mcf) 1 , <f>° 
and performing addition, the coefficients of x Q , x v ... , x m -i are found to 
vanish, with the result 
{x m m\u(d<f>) m } — \m{m+l).m\u{dch) m ~ l d‘ 1 <ji + (m - l)\du.(dcf> m ) , 
this being due to the theorem in differentiation * that the expression of 
m+1 terms 
d r (ucf> m ) - m<f> . d r {uff> m ~ 1 ) + + 1 )</> 2 . d r (u<f> m ~ 2 ) - .... 
has the values 
0, m\u(d<f>) m , \m{m 4- l).w ! uid^^d 2 ^ + (m 4- 1 ) ! du . {dcj> m ) 
according as r<m, —m, or =m + 1. An alternative value for x m is thus 
found, namely, 
\m(m + 1) - - 4- (m + 1 )— i.e. d log [w m+1 (cZ<£y m(m+) ] ; 
d<f> u 
and from the two values it follows that 
A = u m+1 (d<f>) im{m+1) x a constant, 
the constant being determined to be 2[ ± 1°2 1 3 2 . . . (m+ l) m ], or 1! 2!3 ! . . . n ! 
* Attributed in part to Lexell (1772) and to Arbogast (1800). 
