On a Method of Checking Calculations. 57 



place in the ordinary manner, even when the unitate of the 

 subtrahend is greater than that of the minuend. The dealing 

 with fractions is more difficult if our unitates are always to be 

 whole numbers ; but this need not be the case if we take care 

 to preserve the fractional form throughout. The unitates of 



the form U 9 -, however, have U 9 i = 5, U 9 i = 7, U 9 £=2, 



U 9 | = 4, and U 9 | — 8 : these may be formed into the follow- 

 ing Table : — 



1 



xi,- 1 . 



X 



X 



i 



5 



i 



7 



JL 



5 



2 



1 



7 



4 



i 



8 



This leaves U 9 J, U 9 -J-, and U 9 J still fractional because irre- 

 ducible ; and the cases in which they occur must be allowed to 

 retain their fractional form throughout. 



The probability of correctness in the checking by U 9 N (n 

 being the number that furnishes the unitate or remainder), is 

 interfered with by the following peculiarities : — 1. There is 

 no check upon the number of digits in N. 2. Nor upon the 

 place of any particular figure in n. 3. Nor upon the figures 

 themselves, if they be either 9 or 0, or if their sum = 9 or anv 

 multiple of 9. 



Moreover, as evident above, the number of reciprocals or 

 fractions which can be expressed in whole numbers by U 9 N is 

 limited to 66'6 per cent. In 100 there are 33 fractional uni- 

 tates to reciprocals. 



If the function U n N be made to do duty for U 9 N (that is 

 if we cast out the elevens instead of the nines), we gain the 

 following advantages : — 1. There is a check upon the number 

 of digits in N ; for they must be either odd or even, and the 

 addition of a digit to, or its elision from, N, alters the value of 

 U n N. This results from the method of determining U n N by 

 unitation ; for the formula is 



U n N = + «5 - a± + a s — a 2 + a, 



