ALGEBRAIC FORMULA. 285 



proceed in the next place to remark, that although in general 

 the quantity R be expreffed by an infinite feries, yet there are 

 certain values of the arch (p, fuch, that the feries will terminate ; 

 and when this happens, the limiting arch 6 can be found exadl- " 

 ly ; fo that in each of thefe cafes, we obtain a finite equation ex- 

 preffing a relation between the whole elliptic quadrant, and a 

 particular arch of it. 



For we have already obferved, (Art! 5.)j that while <p increa- 



fes from o to a quadrant, or - ; each of the arches <p\ (p", (p'", 

 &c. increafes from, o to a quadrant ; fo that when <p :=-, then 



fin 2(p , fin 4.(f/', fin Sep"', &c. are each 1= o, and 0=:'^ -^ now in the 

 very fame way it appears, that while 29' increafes from o to 

 7, each of the arches 2<t)''\ 2<b"', &c. increafes from o to -: fo ' 

 that, in this cafe, fin 2ip' — i, and ^va.\<p"y fin 8ip"', and all that 

 follow are each =: c ; hence alfo the limiting arch 6 — ""-•■> and 

 to determine the vaKie of ^, we have this- equation — ^" ^^ 



= fin 2(^' =: r, from. which we find cof 2(p =: — e'. 



15. Reasoning in, this way, we find that the feries R will 

 confift of a finite number of terms, and the limit 6 be exa(5tly 

 aflignable in innumerable other cafes, namely, when any one o£ 

 the follotving feries of equations takes place : 



cof 2(p =z — e\ the limit 8 being then = -, 

 cof4(p'zr — f^, - - . |, 



co{%<p"zz — e'\ - - - 1L 



16' 



&c. &c. 



And fince, by means of the feries of equations given in Ar- 

 ticle 1 1, we can determine f, fo that each of the above equa- 

 V0L.V.-P.II. Oo tions 



