ES&3RSB55S8HRE 
925] 
213 
925. 
ON WAKING’S FORMULA FOR THE SUM OF THE mth POWERS 
OF THE ROOTS OF AN EQUATION. 
[From the Messenger of Mathematics, vol. xxi. (1892), pp. 133—137.] 
The formula in question, Prob. I. of Waring’s Meditationes Algebraicce, Cambridge, 
1782, making therein a slight change of notation, is as follows: viz. the equation being 
then we have 
x n + bx n ~ l + cx n ~- + dx n ~ s + ... = 0, 
— me b m ~ 2 
+ md b m ~ 3 
— me') 
+ m.h' 
— m.m — Q. cf 
— m . m — 6 . de 
>b r 
h m- 4 
+ \m .m—S.c 2 j 
+ mf) 
yb m ~ 5 
— m . m — 4 . cd ) 
— m.g 
+ m ,m — 5 . ce 
+ \ m. m — 5. d? 
— \m. m — 4. m — 5 . c 8 , 
yb m ~ 6 
+ \m. m — 5 . m — 6 . c 2 d j 
+ m .i 
+ m .m — 7 . eg 
+ m .m - 7 . df 
+ \m . m — 7 . e 2 
— \m. m — 6. m — 7 . ere 
— | m. m — 6 . m — 7 . cd 2 
+ f^m . m —5.m—Q.m —7. c 4 , 
+ &c., 
- 
where, reckoning the weights of b, c, d, e, ... as 1, 2, 3, 4, ..., respectively, the several 
terms are all the terms of the weight m, or (what is the same thing) in the