526 Mr. W. H. Walenn on Unitation. 



ties, gives —1, showing that each coefficient is — ( — 1)=+1. 

 Other cases may be proved by similar means. When 999 is 

 substituted for 8 in the above form, the coefficients, reduced 

 oy unitation, are also = + 1. 



The following example illustrates this property in regard to 

 the divisors 111 and 999, the given number being the lesser 

 of the two. 



5 8 2 12 3 1-1 2 3 5 8 2 1=4585410. 



17. Some years ago Mr. Henry Butter, of educational cele- 

 brity, showed the writer a property of the decimal equivalent to the 

 reciprocal of 1 9. The right-hand figure of the recurring period 

 of this decimal is 1 ; and the whole period may be constructed 

 from the right-hand end by using the multiplier 2 to obtain the 

 next figure, towards the left, from the previous one, adding in the 

 tens' digit that may be carried from the previous product. The 

 decimal equivalent thus produced is 



•^ = •052631 578947368421. 

 Considering this property with reference to unitation, and be- 

 lieving the case not to be an isolated one, the writer has 

 ascertained that certain decimal fractions can be formed from 

 the right-hand end by means of the coefficients of certain systems 

 of unitates. 



Those decimal equivalents of reciprocals which have the ter- 

 minal figure of their period equal to 1 (the corresponding reci- 

 procal being of the form ) are related to the coefficients 



of unitates, so that the multiplier (a) necessary for their forma- 

 tion is dependent upon the (10 — a) system of unitates. The 

 following example shows this point. 



Take ^ = '025641. Let c v c if c 3 , c 4 , &c. be the coefficients 

 of the function V 6 x (the series of powers of 4), then 



l = c } 

 40 = 10^ 2 



2 

 102 



1600=10% 

 64000 = 10% 



560000 = 10% 

 400000 = 10% 



025641 



and the recurring figures of ■£§ are given by the six right-hand 

 figures of the formula 10% + 10%+10% + 10% + 10c 2 -}- c. 



Since j = J-g, tV^TTT* & c -> * ms ^ aw * s extensible to recipro- 

 cals of the form — -; those of the forms -^ and — 



10^—3 10/* — 10ft — 9 





