QUADRATURE of the CONIC SECTION'S, &c. 323 



Now we have found, that one factor of this expreffion, viz. 



1 

 n (x — 1) 



•r 



— 3 cannot exceed the logarithm of x j with re- 



n . in - 



x m{b — 1) 



1 1 



,i n 



fpect to the other factor b x — 1, fince it appears from the 



b— 1 



firft of the four formulae (/3), (Art. 53.), that b < 1 -f- 



m > 



1 1 r 



and x n < 1 -f- — ~ — - $ therefore, multiplying b m x" < (H J 



(1 -J ^— ), and hence 



\ m J 



f 7-'i < £=ii + izzi + (^-1) (^-i) t 



m n m n 



Now as we may conceive m and /; to be as great as we pleafe, 

 it is evident that this quantity, which exceeds the factor 

 1 1 



b x — 1, may be fmaller than any affignable quantity ', there- 

 fore the product: of the two factors, or the difference between 

 the limits to the value of log x, may, by taking m and n fuffi- 

 ciently great, be lefs than any affignable quantity. 



Upon the whole, then, it appears, that the logarithm of x is 

 a limit to which the two quantities 



1 r 



n [x — 1 ) t n (x - — 1 ) 



1 ,'« 



_ b 



it' 1 



x m(b — 1) m {b" — 1 ) 



Vol VI.— P. II. S s continually 



