382 Proceedings of Royal Society of Edinburgh. [skss. 
preceding year by Prouhet is his own and interesting. Denoting 
the equation whose roots are a x , a 2 , . . . , a n by 
x n -p-^x 71-1 +p 2 x n ~ 2 - = 0 
and the difference-product of the roots by II , he multiplies both 
sides of the identity 
(x-a 1 )(x-a 2 ) . . . ( x-a n ) = x n -p^ -1 +p 2 x n ~ 2 - .... 
by II ; and as the result on the left-hand side is evidently * the 
difference-product of a l , a 2 , . . . , a n , x l he obtains 
| a\a\ . . . <"V| = (x n - f > 1 x n ~ 1 + )n. 
It only remains then to equate like powers of x and there results 
| a\a\a\ .... <;X I = ^n, 
I « • • • I - ftH, 
| a\a\a\ .... I = P»H. 
He points out also that as an alternative to this we may begin with 
\a\a\a\ . . . a r f 1 x n \, express it as a determinant of the next 
lower order, remove the factors (x - oq) , (x - a 2 ) , . . . , (x - a n ) , 
change the product of these into x n —px n ~ x 4- . . . . , and then 
•equate coefficients of like powers of x as before. 
Multiplying again by n he has of course 
| a\a\a \ . . . | . n = (x n ~p- i x n ~ 1 + . . . )n 2 , 
and by changing n on the left into 
1 
1 
1 
. . 1 
0 
«2 
a s . . 
. • a n 
0 
a\ ‘ 
2 
a 2 
a\ 
• • < 
0 
n — 1 
71—1 
a 2 
n — 1 
a, • • 
■ ■ 
0 
0 
0 
0 
. . 0 
1 
and twice using the multiplication-theorem there is obtained 
See footnote to page 365. 
