Mr. R. Templeton on Fractions for the Value of M. 437 

 Let #=3 in equation D vi , then 



4 /11\ 3 10 2 ll 2 - 1 



m 



3 VIO/ 10 2 -1 ll 2 ? 



and if # = 4 in equation ~D UI) we have 



5_/ny / 1Q 2 V 9 2 

 4~\10/ \10 2 -1/ *9 2 -l ; 



so also if x = 5 in equation D /p 



6 _/ll\ 2 11 2 -1 

 5~VlO/ 1L 2 



Substituting these values, we find 



>°=(sr ©"•©■'»' 



11 _ /81V / 656Q V / 1680 V 440 

 10~V80/ \'656l/ 



\121/ 

 In equation A y// let # = 10, and we have 



A 



U681/ 441 



Also in equation B y let nx=%0, ft = 40, x — 2, and we get 



81_121 243 

 80 ~~ 120 " 242' 



Applying these values, we have finally 



/121V 82 /243V 88 /6560\9 2 /1680y 6 /440\ 23 /100V 1 

 \120/ V242/ V6561/ \1681/ ' \4A\) '(,99/ ' 



The number 10 is thus resolved into six fractions excellently 

 suited for computation. 



Another set may be evolved as follows. 



In equation A //y let x — \, then 



l + l_ _/9\ 8 /80\ 4 /25\ 2 8. 

 "T"-^~W A8i; *V24/ 

 and therefore 



*-©"■ ©"• © "■ 



But by equation D,, 



5_ /9\ 2 80 

 4 " \8/ 81 ; 



9' 



consequently 



'25\« 

 ,24/ ' 



