gUJDRATURE of the CONIC SECTIONS, &c. 287 



That is, putting S for To-f 2 ) + T(m + 3 ) + T(« + 4 ) -f, &c. 



S< i [T(m + i) + S], and hence S < — T(« + i> 



Thus it appears, that the fum of all the terms following any 



term, is lefs than — of that term. 

 15 



20. As to the other limit, it muft be the fame as the like li- 

 mit of our firft feries, on account of their having the fame li- 

 mit to their correfponding rates of convergency. That is, 

 putting S to denote as above, then 



S :> T( W +l) rp . 



■L [m) 1 [m+ I) 



nr c> I 1* _ T( w ) — 16 T( w+ i) T 



15 15 V.l(m) l(« + i)J 



21. It yet remains for us to confider how the feries of quan- 

 tities — ^ — r~, — ; tt~ 9 & c ' are to be found. Now this 



1 + cof a 1 + col ~ a 



may be done, either by computing the colines of the feries of 



arches a.- a. - a, - a, &c. one from another by means of the 

 248 



formula cof - A= V * "*" coi , and thence computing the fe- 



2 2 



cc ■ jq.« I — cof4-tf i — co{±a Q ~ 



ries or fractions — ; ^-f — , — ; 7*7—, &c. Or we may 



1 + cof -i- a 1 + cof ~ a J 



compute each fra&ion at once from that which precedes it, by 

 a formula which may be thus inveftigated : 



Put 



