Mr. W. H. Walenn on Unitation. 547 



into definite language : — The decimal equivalents of reciprocals 

 of the form in ^ may be constructed, without the aid of di- 

 vision, from the right-hand end of the period, solely by means 

 of the multiplier a, this multiplier being used to obtain the 

 next figure towards the left from the previous one as a multi- 

 plicand, and adding in the tens' digit that may be carried from 

 the previous product. 



Expanding the formula in . , by division, it becomes 

 11 11,1 1 



10a ' (10a) 2 ' (10a) 3 ' ' * ' ' (10a)"~ 2 ' (10a)*- 1 ' (10a)* ? 



n being the number of figures in the recurring period of the 

 decimal. By art. 18, it has been proved that the last term, 



uamel " v (IKF is L If (W* = 1 ' theu 



1 10a _ 



(10a)*- L ~(10a)«~ iUa; 



and the series (commencing from the right-hand term) becomes 

 ... +(10a) 4 + (10a) 3 + (10a) 2 + 10a + l. 



The above theorem is thereby proved in relation to reciprocals 

 that have two figures in the denominator, a being the multi- 

 plier therein mentioned. By exactly similar reasoning, the 

 theorem may be extended to reciprocals having more than two 

 figures in the denominator. 



20. In regard to the remainders to the division that forms 



the reciprocal =-» 1 ? by taking an example, the relation of 



the unitates (to the base 10) of these remainders to the figures 

 in the quotient will be distinctly seen to be as follows : — 



39)l-000000(-62564i 

 78 



220 

 195 



250 

 234 



160 

 156 



4 



9 



1 



202 



