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 exact- 

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

 prefling a relation between the whole elliptic quadrant, and a 

 particular arch of it. 



For we have already obferved (Art! 5.)* 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 



fm 2<p', fin 4<p // , fin 8<p"\ &c. are each = o, and & zz - ^ now in the 

 very fame way it appears, that while 2<p' increafes from to 

 -, each of the arches 2<p" f 2<p'", &c. increafes from o to -; fo 

 that, in this cafe, fin 2<p' zz 1, and £m/\<p% fin 8<p"', and all that 

 follow are each r: c ; hence alfo the limiting arch 6 — -\ and 



to determine the value of <p, we have this- equation 



= fin 2$' zz 1, 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 be exactly 

 affignable in innumerable other cafes, namely, when any one of 

 the following feries of equations takes place : 



cof 2<pzz — e\ the limit & being then rz ~ t 



cof 4<p' — — e\ 



cof 8<p* = — *", 



4 



8> 

 &C. &C. 



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

 ticle 11, we can determine <p, fo that each of the above equa- 

 Vol, V. —P. II. Oo tions 



