x+1 



X 



Mr. R. Templeton on Fractions for the Value o/M. 435 



ever, very swift convergence ; but the work for both is far less 

 onerous than the computation of the terms of the original frac- 

 tion. The above may be put under a slightly different form : 



x + \ _ / 237+i y (2# + l) 2 -l . 



w ~\ %x )' {2x + l) 8 '' ' ' '" 



and by substitution, 



_ / 4a? + l \ 4 / (4r + l) 8 -l \ 8 (2*? + l) 2 -l 



\ 4x J \ (4^-fl) 2 / (2,r+l) 2 ' ' ' * ' " 



_/ 8a? + l \ 8 / (8a7+l) 4 -l \ 4 / (4^ + l) 2 -l \ 2 (2a7 + l) 8 -l 

 \ 8# /'\ (8^+l) 4 / \ (4^+l) 8 / (2* + l) 2 A/ " 

 &c. &c. &c. 



The equation A, is only a particular value of a more general 

 expression, in which sc receives a small multiplier n (positive and 

 integer), — practically, however, with this disadvantage, that every 

 unit of which n is composed indicates the requirement of another 

 fractional factor : some such expressions are 



_( nx+l \ n / (na? + l) 8 — I V- 1 / (wa? + 2) a — l \ n - a / (wa? + 3) 8 — l y 3 

 'U y\ (nx+lf ) A (nx + 2f / A {nx + 3) 2 / '^' 



(nx + 1) 2 (nx+2\ n . { (rix + 2) 2 — l y-^ / (^ + 3) 8 — i y~ 3 

 "(nar+l) 8 -l* W+l/ A (^ + 2) 2 ) \ (rc^ + 3) 2 ) C ' 



[nx + \f / jnx + 2) 2 y / nx + S \ 4 % / (na? +3) 8 -i y- s fr 

 "(n»+l) 8 -l \(na? + 2) 8 -l/ # W + 2/ \ (/u> + 3) 2 / ' C ' 



(y^ + l) 2 . / (nar + 2) 8 \ 8 , / jnx + 3)* \ 3 . / tw + 4 y „ 

 - (nx + l) 2 -l\{nx + 2) 2 -l) V(nar + 3) 8 -l/ W + 3/ ' C * 

 &c. &c. &c. 



It is sometimes advantageous to change a fraction into another 

 differing little from it, but whose denominator shall be increased 

 or diminished by an aliquot part of the denominator of the ori- 

 ginal fraction. The form for this is 



nx + 1 _nx + n+l nx 2 + nx + x + 1 

 nx nx + n nx 2 -\-nx-\-x 



_ nx—n + 1 ^nx^—nx + x + l 

 nx—n nx 2 — nx-\-x 



nx — 1 nx + n — 1 nx 2 -\-nx—x—l 



B,. 



B,, 



B ///- 



nx nx + n nx' + nx — x 



Useful particular values for more immediate reference may be 



