Mr. R. Templeton on Fractions for the Value of M. 439 



And if 7i#=404, ?i = 101, x =4, 



405_401 40500 

 404 ~ 400 ' 40501 ' 



and from equation A ;/ , 



101 _ /401V /160800X 2 40400 

 100 " V400/ ' V160801/ ' 40401* 

 But 



401_501 2005 



400"" 500*2004* 



and if ^#=2004, #=501, and n=4 in equation B /;/ , 



2005 _ 2001 1002500 

 2004 ~ 2000 ' 1002501 ' 



Finally, from equation A /y , 



501 _ /2001V /4004000V 1002000 

 500"" V2000/ ' V4004001/ ' 1002001* 



Substituting these values, 



/2001\ 4554 /2500\ /2401V /4004000V 840 

 \20007 ' \2499/ • V2400/ ' V4004001/ 



/1002000\ 920 / 1002500 \ 874 /160800\ 432 / 40400 y c 

 * V1002001/ f V1002501/ " U60801/ * \404017 



/10200V 03 /251000V 3 /12800V 3 /40500V 

 * V 10201/ ' V251001/ ' \128bl/ " \40501/ ' 



We thus see that by assuming different forms of fractions in 

 the preliminary steps, and carefully examining the results, very 

 superior sets of fractions may be hit on, and the modulus com- 

 puted, to any amount of precision that may be fixed on, with ease, 

 not perhaps so much from saving of figures as from the vast 

 amount of relief from mental labour that such efficient divisors 

 secure. 



