QUADRATURE of the CONIC SECTIONS, &c. 335 



f 



X 



log 2 X 



= ^ 



+ -i 



(x—iy ^ 12 



1 X'— -i , 1 X"— 1 



i± % X' 



I X"'— T 



4 2 X'+i + 4 3 X'+i + 4 4 X'"+i * ' * 



+ T {m ) + T (wj + i) + R} 5 



and here T( m ), T( m + i), are put for any two fucceffive terms of 

 the feries, and R for the fum of all the following terms : And 



in every cafe R is greater than — T\ w + i), but lefs than 



I rp , l6T(m+i) Ttni) rp, 



64. From the analogy of the two formulae 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.). 



Let x be any number, and X, X', X', X"', &c. a feries of 

 quantities formed from x, and one another, as fpecified in the 

 beginning of the laft article. Then 



log 4 * 



