QUADRATURE of the CONIC SECTIONS, &c. 329 



and this is the firft feries which I propofe to inveftigate for the 

 calculation of logarithms. 



59. The feries juft now found agreeing exactly in its form 

 with our firft feries for an hyperbolic fector, (Art. 40.), as it 

 ought to do, will of courfe have the fame limits to the rate of 

 its convergency, and to the fum of all its terms, following any 

 propofed term. As the latter of thefe have been deduced from 

 the former, in the cafe of the hyperbola, by a procefs purely 

 analytical, and the fame as we have followed in treating of the 

 rectification of the circle, it is not neceflary to repeat their in- 

 veftigation in this place. The limits to the rate of convergen- 

 cy, however, having been made to depend partly upon the na- 

 ture of the curve, it may be proper, in the prefent inquiry, to 

 deduce them entirely from the analytical formula which has 

 been made the balls of the inveftigation. 



Let any three fucceflive terms of the feries of quantities 



j_ ±_ 



x 2 — 1 x* — 1 



— ; , — , &c. be denoted by t, t' and t" ; then it is 



x* -f- 1 # 4 -f- 1 



evident from the formula, (Art. 58.), that the relation of 

 thefe quantities to one another will be expreffed by the equa- 



tions 



6 — i 4- t' 



2 1 , :« 1 „/, 



t 



From the firft of thefe we get 2 t' zz t (1 -j- t' z ), now each of 

 the quantities t, t', &c. being evidently lefs than unity, it fol- 

 lows, that 1 + f' 2 < 2, but > 1, and therefore that 2 t' < 2 t, 



and /' <tj alfo that 2t'> t, and t' > -t. Hence it appears, 



in 



