On the Method of developing an Incommensurahle Fraction. 129 



plied the multiplier; and it is to be observed that each addend 

 is obtained by merely multiplying the preceding addend, and 

 not the whole preceding row. 



In the instance chosen by Colson, the several multipliers 

 which he employs, one after another, happen to be conveni- 

 ently small : it is not so with the fraction — , Treated in the 



way here suggested, the development of this fraction is as 

 follows : — 



JL — 'OS— ^^''^ *^^^ multiplier is 7, and to this we may 

 31 "" 31* keep throughout. 



31 

 _ 2 Here the multiplier 7 may be changed 



~ • • "^^ 31* for 2. 



= •032258,064.516^ 

 31 



= 129032 ^ 



31 



.•.—• = •032258064516129032 — 

 31 31 



By Colson's method, the multipliers for this fraction are 

 7, 18 and 14; and although, in that method, the number of 

 decimals is doubled at every step, yet this is no advantage; 

 for in both methods n new figures require n multiplications: 

 the only point of difference between the two methods is this, 

 namely : — in Colson's method we are to employ the successive 

 multipliers as they arise, whereas in the modification of it, 

 here proposed, we may reject those multipliers which are 

 inconveniently large, and use only the smallest. 



After I had completed my former paper, a vague recollec- 

 tion revived in my mind of having somewhere seen a process, 

 by Colson, having some analogy to the speculations with which 

 I had been occupied ; and I delayed the communication of the 

 paper till I had sought for information on the matter from a 

 distinguished mathematical friend, who was not able, however, 

 to call anything to his remembrance, in connexion with Col- 

 son's name, at all allied to what I had been doing. I have 

 since consulted Newton's Fluxions, and have there discovered 

 what I must have read many years ago ; and the present sup- 

 plementary communication is the result of my examination 

 of Colson's rule. This rule admits of an application and an 

 extension of more general utility than the author seems to 

 have conceived ; and it is chiefly to show this, that I submit 



Phil. Mag. S. 3. Vol. 36. No. 241 . Feb. 1850. K 



