QUADRATURE of the CONIC SECTIONS, &c. 337 



here m is put for any integer power of 2. Let the product of 

 the correiponding fides of thefe equations be now taken, and 

 the common factors rejected, and the remit will be 



, ™ v x 2 -f- 1 x A 4- 1 x 8 4- I x m + I 

 x — i—m(x — 1) ■ ■ — : — . . . ■ — 1 



and hence 



1 2 



m (x m — 1) = (x — 1) 



' X -f* I 



This equation holds true, m being any power of 2 whatever. 

 Let us, however, conceive it indefinitely great. Then the 



1 

 number of factors will become infinite, and m (x m — 1) will 



become Nap. log * (Art. 57.). Therefore, 



2222 



Nap.log#-(> — 1) -j- -7 j ^ , &c. 



x 3 - -f 1 x A + 1 x* -f- 1 x xZ -\-\ 



ad infinitum. 

 The product of any finite number of thefe factors being al- 



ways a function of this form m (x' a — 1) will of courfe be great- 



1 1 



cr than log x, (Art. 54.). However, the function ~ m {x' — 1) 



m 



X 



or m I 1 — — V being in like manner expanded into an infi- 



rm 1 — — 1, being in li 



x n ) 



nite 



