316 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 ;'-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. Ir 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 abis, ab+s, ab4s, &c. from the abfcifla correfponding 
to the whole fector, and from one another by the known for- 
mula 
abS+1 
ab 4S =A/ 3 
abts—1 abis—I &e 
and thence the values of the quanties ——2—~—_—, —_4 = 
q ~ abes+a abis+r 
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 
abS—r ab>S—rI j 3 re 
== 7, as = 7 = 
ab SE , and abis Seok , then we have ab § er, 
and eae ee +3‘ we have alfo ab Six taht and 
fince by the nature of the hyperbola ab}S= NA Siege 
therefore 
