148 
b x d +f* m + — b x d Ar j * m x e 
j m 
Dr. Waring on 
x 4. &c. = P ; extract the root 
4- c x d +/% 
y/h + k7- B + — + &C. = Q ; add the three quantities P, Q, 
V 
and x~ 2 + a contained under the fame root \/ together, and the 
feries refulting, whofe terms are conftituced according to the 
I 
dimenfions of x 9 will be x~ z + (a + b x d+f* m + fc = A) 
1 - — m 
( Lb xT+ 7 l~ x e + cxd = B)* + &c.; of which ex- 
V m J 
tra£t the cube root v 7 , and it will be x~ * + fA** + + &c. 
the root required. 
5. The principal ufe of reducing quantities into feries pro- 
ceeding according to the dimenfions of the variable quantity is 
as before mentioned for finding the area of a curve from its 
ordinate ; or, which correfponds, the integral from its nafcent 
or evanefcent increment ; but the feriefes deduced ffiould con- 
verge, otherwife from them cannot be found the area or integral. 
In the Meditationes Analytic;® a method was firfi: publilhed 
of finding when thefe feries will converge and when not, e. g. 
the feries a -\-bx-\- cx 1 + dx 1 4- &c. = J (A + Bx + Cv +D^ 3 + 
&c.) K xx = P will converge when (v) either affirmative or ne- 
gative is lefs than the leaft root («) of the equation A + Bv + 
Osf -v-Dx 3 + &c.=o, if the roots are poffible. A fimilar rule is 
given when fome of the roots are impoffible. The feries will di- 
verge when * is greater than «, and the cafes are given in which 
it will converge when x =«. The feries defcending according to 
the dimenfions of x will converge when * is greater than the 
greateft root of the equation, &c. Thefe principles are fur- 
