346 Mr. W. H. Walenn on Unitation ; 



strips correspond, according to the system there adopted, to the 

 values cf)= +2K, where x= + 2AK gives half the distance of the 

 strips, while the breadth of the strips depends upon the relation 

 between the constants A and B. 



From the form of the equations (2) and (4) we may see that 

 <f> and fy can only be expressed as functions of x and y by ex- 

 ceedingly complicated serial development. 



XLIV. On Unitation ; a novel Arithmetical Operation. 

 %W. H. Walenn, F.C.S* 



IT has long been known that the remainder to the division of 

 any number by 9 may be determined by dividing the sum 

 of its digits by 9. The following theorem is a generalization and 

 explanation of this property, and establishes a similar operation 

 for all other divisors less than 10. 



The theorem is, that if / be the tens' and u the units' digit of 

 a two-figure number, and 8 be any integer less than 10, then 



10— $t + u 

 has the same remainder to 8 as lOt + u. 



For 10 — 8t + u = \0t + u — St j and the latter expression is only 

 different from lOt + u by an exact number of times 8. 



Further, if the digits be s, t, u, the same is true of 



10-8s+10-8t + u, 



and so on. For, each time 10 occurs as a factor in any term, it 

 must be treated in the way above indicated, and 



10-8 s+10-8t + u = 10-8 10-8 s+10-8t + u. 



These formulae show the means of determining the remainder 

 to any digit without knowledge of any multiple of that digit, 

 inasmuch as the operation shown by the formula 10 — 8t + u only 

 involves multiplication and addition (whenS = 9, addition only) ; 

 instead of dividing the result of the said operation by 8 to obtain 

 the final result, the operation is repeated until only one digit, 

 necessarily one or more units, is obtained. The fitness of regard- 

 ing this method of obtaining the remainder to 8 as an operation 

 totally distinct from that of division, and the fact of single digits 

 only being used, make it convenient to call the said operation 

 " unitation," the remainders being " imitates/' and the divisor 

 the " base." 



In practice, unitation itself may be simplified by commencing 



* Communicated by the Author. 



