QUADRATURE of the CONIC SECTIONS, &c. 335 
: pgietans Spe See 
| (w—1y* wi 12 
zy wu 
ieee ee pe aa et 
ie i X’+1 4° X’+1 its 4 X"+1 °°" 
Lo + Twn) + Tein + RE 
and here T(m), T(m+z1), are put for any two fucceflive terms of 
the feries,-and R for the fum of all the following terms: And 
in every cafeR is greater than # T(m+41), but lefs than 
I 16 T(m+1) — T(m) 
=" T m <= - PANT el Ae ee m I)}e 
ue eet Ted) 
64. From the analogy of the two formule from which we 
have deduced the feries for the rectification of an arch of a 
circle, and for the calculation of logarithms, it is eafy to infer 
that there will be correfponding feries for the refolution of 
each of thefe problems. And as the two preceding feries for 
a logarithm have been inveftigated in the very fame way as the 
firft two feries for an arch of a circle, fo, by proceeding exact- 
ly as in the inveftigation of the third and fourth feries for the 
circle, we may obtain a third and fourth feries for a logarithm. 
The mode of deduction, then, being the fame in both cafes, 
and alfo fufficiently evident, I fhall fimply ftate the refult of 
the inveftigation of a feries for logarithms which is analogous 
to our fourth feries for an arch of a circle, (Art. 28.). 
Ler x be any number, and X, X’, X’, X”, &c. a feries of 
quantities formed from x, and one another, as f{pecified in the 
beginning of the laft article. Then 
I 
logts — 
