i 68 Dr. W aring on the 
obferved, in the before-mentioned Book, that (if the common 
divifor be {a - A)) it will once only admit of 5, 4, c, . . r equal 
roots ; if it be a quadratic, then it will twice admit of thole 
equal roots ; and fo on. 
4. If the roots of the equation of the leaf! dimenfions be 
fubftituted for a in the remaining equations, and each of the 
refulting values of the fame equation be multiplied into each 
other, there will refult the r ~ 1 equations of condition : and: 
the fame may be deduced alfo from the feveral equations con- 
jointly. 
The equations of conditions found by the firft method, if 
the divifions were not properly inflituted, may admit of more 
rational divifors than neceflary, of which tome are the equations 
of conditions required, 
, ' 2 . 
1. In the year 1776, I publifhed in the Meditationes Ana- 
lyticae a new method of differences for the refolution of the 
following problem. 
Given the fums of a fwiftly converging feries ax + bx z + cx 3 
-p dx 4 -f &c., when the values of x are refpedtively tt, p, g , &c. > 
to find the fum of the feries when x is t, that is, given 
S7Z" ~ a 7? -p ^7r m \ m -C'7r -p dor 4 -p ecc, Sp == -}- bp cp -f- &c. , So* ~ 
acr + bed -p cor 3 -p &c. &c.- ; to find St = ar -P bd -p cd -P &c. 
To refolve this problem I multiplied the quantities, S7r, Sp, 
S a-, &c. refpedively into unknown coefficients oc 9 @ 9 y , &c. 
and there refulted 
