Mr. W. II. Walenn on Unitation. 525 



into 152 for instance), and the lesser number subtracted from 

 the greater, the remainder would be exactly divisible by 9. The 

 statement was then extended to deriving the second number from 

 the first by transposing the figures of the first in any order. 

 The proof of this theorem by imitation presents some interesting 

 points, inasmuch as it shows how this property of numbers may 

 be expanded still further. 



If B be the number thus derived from the given number A, 

 then U 9 A = U 9 B. For A and B have the same digits, and the 

 order of their addition does not affect the unitate of either of the 



numbers. Therefore U 9 A — U 9 B=0 or 9; or — - — (— 77 — if 



B be greater than A) gives no remainder. 



Further, the derived number may have either noughts, or 

 nines, or multiples of nine in addition to or in defect of those 

 in the original number, and the theorem will still be true. 

 For example, 



105063259723-153362752 = 104909896971, and 

 U 9 (7-7)=U(16-7) = 9, 



which is the unitate of 104909898971. 



16. The derived number may be formed from the given num- 

 ber by other methods, so that the remainder to the subtraction 

 may have other properties ; unitation suggests these properties 

 and furnishes proofs of the same. If dots or marks be made at 

 equal intervals over the digits of the given number, or a part of 

 the given number, and (having regard to the relative position of 

 the marked digits in the number) if the marked digits be trans- 

 posed among themselves, the remainder will be exactly divisible 

 by some number composed entirely of ones, as 1111 for instance, 

 also by some other number composed entirely of nines, as 9999 

 for instance, the number of ones or of nines in the divisor being 

 tf 4-1, if a be the number of intervals. If the marks be placed 

 over the 1st, 3rd, 5th, &c. digits (leaving one interval), the re- 

 mainder will be divisible by 99 and 11, if over the 1st, 5th, 

 9th, 13th, &c. digits (leaving three intervals), the divisors will 

 be 1111 and 9999, and so on. 



For if the unitation formula (see article 11) be arranged so 

 as to assume the form 



(io~a) 3 - 3 /+(io-a) 2 - 3 o+(io-8)v+«, 



and 111 substituted therein for S, each term (when unitated to 

 the base 111) will be found to have +1 for its coefficient. In 

 this case the substitution gives negative values to every second 

 term commencing with the second from the right hand ; but the 

 subsequent unitation of these values, taken as positive quanti- 



