436 Mr. R. Templeton on Fractions for the Value ofM. 



selected from the above, and will be made use of in the examples 

 which follow. 



x + 1 30 x 2 — 1 r < 



X X-\-\ X 2 



x + 1 fax + l\* (2^ + l) 2 ' 



D, 



1 __ / 2a? + l V 



a? ""\ 2a; / (2ar+l) 



_/2tf + 2\ 2 (2* + 1) 2 -p. 



~W + 1/ "(2ar + l)«-l »' 



_ /2a? + 3\ 8 . / (3* + 2) 8 V . (2^ + l) 2 



~\2a? + 2/ \(2« + 2) 9 -l/ (2a7+I) 2 -l' ' '"' 



_/ gg + l \ 8 /(3a? + l) a -l \ g (3*+2) 8 -l 

 ~V~3i7~/'\ (3a? + l) 2 /' (3a? + l) 8 ' iv> 



- 'j te + V f . (3*?+l) 2 t (3^ + 2) 2 ~l 

 ~\3ar+l/" (3a? + l) 8 -l' (3* + 2) 8 * ' v " 



_ /3a?+3y . (3^ + l) 8 / (3^ + 2) 2 \ 8 

 * "\3fl?+2/ ' (3a? + l) 8 -l \(3a? + 2) 8 --l/ ' vi * 



ar-1 / 2g-i y (2ff-l) 2 -l 

 x ~~\ 2x ) ' (Sfcw-1) 8 " 



x _ / 2af + i y / 4a? 9 V (2a?- 1) 2 

 a?-l \ 2a? / W 2 -!/ '(2a?-l) 2 --l ' ' D 



&c. &c. &c. 



In the examples the mode of procedure has been to make use 

 of one or other of these expressions, as it may happen to suit 

 itself to our immediate purpose,' — that purpose being to select, 

 with careful discrimination in the selection, a promising set of 

 primary equations to develope ; developing these with a view to 

 coalescing as many fractions as possible into others with larger 

 denominators, at the same time trying to secure as many small 

 digits or ciphers as possible in the denominators, so as to render 

 them easy divisors, and, where we fail in this, so large as to have 

 but few terms to compute. We may then proceed as follows : 



Vll* 



Vlll* 



6_6 5 4 

 ^~3""5'4V 



5 



and since 2 3 . 7 '=10, 

 4 



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