RECTIFICATION of tie ELLIPSIS, &c. 185 

 It is manifefl we may proceed in this manner as far as we 



pleafe, and that, if we put c'" =. - — ^^' " . ; c"" — - / 



and fo on, we fhall have the value of A in an infinite produdl, 

 A r= (i + c') X (i + O X (i + c'") (i + c'"') X &c, the 

 quantities c\ c'\ c"\ c"", &c. converging very rapidly. 



Nothing more feems to be wiflied for, with regard to the 

 computation of the quantity A : fince we can, by methods fuf- 

 ficiently fimple, exhibit the value of it in feries that fliall con- 

 verge as fall as we pleafe. By a fimilar mode of reafoning, I 



find the feries i — i- y' + ifi! y* _ lllllil y» + &c. 



(which occurs in determining the time of a body's defcent in 



the arch of a circle), rr (i — c) X {i -{- ^ c^ -f- VA c* -f- 



^-^^^^^^c^ + &c.) where c = ^i±l! : fo that the fumma- 



2.4.0 / ^i ^y» + I 



tion of this feries alfo is accomplifhed by the method above. 

 :o explain the method of computlm 



= A + B cof<p + C cof 2<p + &c. 

 1 there refults 

 rz B -f (2A + C) cof (p- -f &c. 



I HAVE now only to explain the method of computing B. For 



, . - r T r .Oi;D:j.i.V d 



this purpole 1 relume, ,^ 



I 



Multiply by 2 cof <p, and there refults 



V^ I + C* 2C COi'cp 



whence it is manifefl that the whole fluent 



I ■ ^'^° 'P ^ , , -when (p zz ar, is equal to B X w. 



From the preceding invefligation we have ^ 



v' I + c^ — 2c cof 9 



=f=, and cof <p := c fin 'J' + cof ^I' '/i — c^ fin ^i^. 



v^ I — c^ fin 'g.' 



Voi,. IV. Z whence 



