RECriFICAtlON of the ELLIPSIS, &c. 187 



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

 A' and B', we fliall have 



B = ..(i+0(A' + 5:) 



B' = .'.(i+0(A"+ t) 



B" = c'.ii + c"') (A'" + ~) Sec. : 



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

 &c. we have the following values of B : 



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



B=.x(i+j+^4)(i+o(i+o(i+oA"'+^,-^-4(^+^'> 



(i+0(i + c"')B"'. 



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

 feries that fliall converge as faft as we pleafe. 



As the quantities c\ c", c"', &c. diminifh 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, fince A = [1 + c') {i -^ c") {i + c'") &c. : 



B = . X (i + I + TT+f T'T + S'^-) X A- 



We fhall beft fee the degree of convergency of the quantities 

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



I — ^i — x^ 



rived one from another. Now,, if V r: — — 7~=r, then alfo j' z: 



' IInr1l3W 0:3' + ^"-'' 



— +-g-+g — j- j-g-f- Sec. : whence it is obvious, that in the 



feries of quantities f , c\ c*, &c. the fourth part of the fquare of 



Z 2 any • 



