3 i6 NEW SERIES for the 



that by proceeding, as in the cafe of the circle, to determine 

 limits to the fum of all the terms following any afligned term, 

 we would obtain an analogous refult, namely, that the fum of 

 all the terms following any afligned term is greater than T ' T th of 

 that term, but lefs than a third proportional to the difference 

 of the two terms immediately before it, and the latter of the 

 two. 



47. It now only remains to be confidered, how the numeri- 

 cal values of the terms of the feries are to be found. Now, this 

 may evidently be done by computing the values of the quanti- 

 ties ab 4- s, ab~ j, ab J- s, &c. from the abfcilfa correfponding 

 to the whole fector, and from one another by the known for- 

 mula 



and thence the values of the quanties T ~~ - , 4 s -, &c. 



ab 4- s 4" 1 ab ~ s + 1 



Or we may deduce each of thefe from that which precedes 

 it, by a formula analogous to that found at Art. 21. in the cafe 

 of the circle, and which may be inveftigated as follows. Let 



- — t, and a ^ -|~- ^- — t\ then we have ab S = ii- , 



abS 



ab S + 1 ~ ' abfS+i 



and ~±— — ; we have alfo ab-i-Szr-?— t — ; and 



2 1 — t 1 — t 



lince by the nature of the hyperbola abjS~v r-— $ 



therefore 



