QUADRATURE of the CONIC SECTIONS, &c. 315 



of thefe inquiries, it appears, that the terms of the feries, un- 

 der its firft form. (Art. -3) are exactly the fquares of the cor- 

 refponding terms of the former feries. under its firft form 

 (Art 36.), lb that the one being written thus, 



I = P — (T (I) + T (2 ) . . . . + T (B) + T( M + I ) -f- T (w + 2) +, &c.) 



the other will be 



- a = F— (T\ I )+T 2 (2 )... + T 2 ( w ) + T> + I )-fr ( « + 2 )+,&c.), 

 s 



and here P and P' are put for the parts of the two expreflions 

 which do not follow the law of the remaining terms, but Tu), 

 T(i), &c. denote the fame quantities in both. Now, as each 

 term in the former feries has been proved to be greater than 

 one-fourth of the term immediately before it (Art. 39 ) each 

 term of the latter muft be greater than one-fixteenth of the 

 term immediately before it ; and this is one limit to the rate of 

 convergency. 



Again, as it has been proved (Art. 40.), that in the 



firft feries T(„-j- 2 ) < — ^ — -, therefore, fquaring, we have 



T 2 (n + 2) < >p" • Now this quantity is a third proportional 



to T 2 («) and T 2 („ + o ', hence it follows, that the greater limit of 

 the rate of convergency in the two feries is the very fame; 

 that is, each term is lefs than a third proportional to the two 

 terms immediately before it. 



As thefe limits to the rate of convergency differ from 

 thofe of our fecond feries for an arch of a circle (Art. 18.), 

 only by the leller limit in the one cafe correfponding to the 

 greater in the other, and the contrary, it is fufficiently evident, 



Vol. VI — P. II. Rr that 



