RECriFICJTlON of the ELLIPSIS, Sec. 187 



A", A'", &c. ; B", B'", &c. to denote the correfponding values of 

 A' and B', we fhall have 



B = ..(i+0(A' + v) 



B' = .'.(i+."){A''+ t) 



B" = c". (i + c"') [M" 4- —-) &c. : 



Now, remarking that A' rr (i + c") A"; A" = (i + c") A'", 

 &c. we have the following values of B : 



B = cX(i+^-).(i+0.(i + OA" + H'Ci+0(i + OB". 



B=.x(i+;-+^4)(^+0(i+0(i+OA"+rrT^^+''>- 



And we may proceed in this manner to find the value of B in 

 feries that ihall converge as faft as we pleafe. 



As the quantities c\ c'\ c", &c. diminilh very faft, the feries 

 A', A", M" will approach rapidly to unity, and B', B", B'" will 

 decreafe rapidly to nothing : Hence we have ultimately, 



&c. 



or, fmce A z: (i + C) (i + c') {i + c'") &c. : 



B = . X (i 4- j + f 4^+f T-T + ^^O X ^- 



We {hail beft fee the degree of convergency of the quantities 

 c, c\ c\ 8cc. if we take the infinite feries by which they are de- 



rived one from another. Now, if y = ' T~ / ^ ~r > ^^^^ alfo j^ iz: 



1 -T Y I — X 



I y/l — x^ 



acner. i\ow, irj = 



— + -5 — f- 2 1 + &C. : whence it is obvious, that in the 



4 ' 8 ' 64 ' 128 ' ' 



feries of quantities c, c\ c'^ &c. the fourth part of the fquare of 



Z2 any 



