94 Mr. waring on the General 
To render the folution general, it may not be impro- 
per to fubjoin the fubfequent. 
L E M M A. 
i , Let «, , 6 , y, s, £, Sec. be the refpeciive roots of the 
equation z"— i ; then will <x. m + +y m + + Sec.=o., 
unlefs n=m, or n is a divifor of m, in which cafe u m + 
( 3 m +y m +S m +£ m + Sec. - n . 
a. The fum of all quantities of the following kind 
9 *P+«rp m +zY+oi r y m +l 3 Y+fiY + K m d r + &c. wili be 
= o ; unlefs n be either equal to, or a divifor of m +r f 
in which cafe the fum above-mentioned will be = — n ; 
except n be either equal to m or r, or a divifor of them, 
in which cafe the fum will be n l ~n\ but if m=r t then 
in the former cafe will the above-mentioned fum = — -» 
2 
and in the latter = 
-2 
3. The fum of all quantities of this kind z m ( 5 r y s o t Sec. + 
&c * + 8cc. + cc r @y m V See. + 8c c. will be 
— o, unlefs n be either equal to r+m+.s + t See. or a di- 
vifor of it. 
Let 7t be the number of indices rn, r, s, t, 8cc. and 
n be either equal to r + s+t + 8ec. or a divifor of it, 
but n be neither equal to, nor a divifor of the fum of 
any two, three, four, . . . 71-3, 7T- 2 or z - 1 of the above- 
mentioned 
