RECTIFICATION of tie ELLIPSIS, &c. 185 

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



— I — y/T-^"' . ,„, _ I — s/i— C ^ 



+ v/»— c"' ^ I + v/i 



pleafe, and that, if we put c'" zz - 



and fo on, we ftiall have the value of A in an infinite produ(5t, 

 A = (i + r') X (i H- c") X (i + t'") (i + c"") X &c, the 

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



Nothing more feems to be wifhed 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 fhall con- 

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



find the feries i — i-o.' +iliil y^ _ ^\'K't y' + &c. 



2"'' ' 2'. 4' 2..'). 6^ ' 



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

 the arch of a circle), — (i — <r) X (i -f- ^ ^^ -\ r-^ ^^ "f 



\^^-^-^c^ + &c.) where c = ^l±2l — - : fo that the fumma- 



2.4.0 / ^i + yi ^ I 



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



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

 this purpofe I refume, 



,- / r: A -f B cof(2> + C cof 2® + &c. 



Multiply by 2 cof (p, and there refults 



; ^V°^^ ' , = B + (2A + C) cofcp + &c. 

 whence it is manifeft that the whole fluent 



r 2C0 (px <p ^ when © zr ra-, is equal to B X zsr. 



From the preceding inveftigation we have 



V' I + c^ — 2c cof ^ 



=, and cof <p ir c fin ^'^ -{- cof 4' ■^i — C" fin ^^--j 



Vol. 1Y. Z whence 



