[ 166 3 
CVII. On the Method of correfpondent Values , &c. By Edward 
Waring,, M, D. F. R. 6\ and Lucafian Profejfor of the Ma . - 
them a tics at Cambridge.. 
Read May 28, 1789. 
1. 
I-.. TN the year 1762 I publifhed a method of finding when 
-L two roots of a given equation x n —px ”" 1 -f qx n ~~ z — ra"“-3 4. 
o are equal, by finding the common divifors of the two 
and 
quantities a n —pa r ‘~~ l + qa n ~ z — &c., 
na 
n — I 
-n— ipa n ~ z 4- 
72 — 2 qa n 3 — &c., and obferved if they admitted only one Ample 
divifor (a — A), then two roots were only equal ; if a quadratic 
(<3 2 — Atf-fB), then two roots of the equation became twice 
equal; if a cubic (V — Arr 4- — C), then two roots became 
thrice equal ; and (o on : or, to exprefs in more general terms 
what follows from the fame principles, if the common divifor 
be a — b r x a — c s x a — d ( x &c., then r+i roots of the given 
equation will be h 9 .*4- 1 roots will be c, /+ 1 will be d 9 &c. ; 
and it immediately follows, from the principles delivered in 
the fecond edition of the fame Book, publifhed in 1 770, that 
to find when r+i, v + i 9 / 4- 1 , &c. roots are refpe&ively equal 
requires r + s + t 9 &c. equations of condition, which are dedu- 
cible from the well known method of finding the common 
divifors of two quantities in this cafe of a n — pa ”- 1 + qa”~~ z - &c., 
na *" 1 — n - \pa n ~ z 4-/2 — 2qa n ~~* — &c, of the terms of their 
remainders, &c. 
In 
