[ 346 ] 



XXXIX. On Unitation. — III. The Unitates of Powers and Roots. 

 By W. H. Walenn, Mem. Phys. Soc* 



IN former papers f it has been shown that if a, b, c,. .. s, t, u 

 be the digits of a given number, and 8 be any integer less 

 than 10, n being the number of digits in the same number, the 

 expression 



{10-S) n - 1 a+ [l0-8) n - 2 b + (10-8) n - 3 c+ ... + (10-S) 2 s 

 + {l0-8)t + u 

 has the same remainder to 8 as the given number has. 



By simple substitution of 8 in this expression, the remainder 

 to 8 may be found, two kinds of operation being necessary for 

 that purpose. The first is the determination of the coefficients 

 (10-S) n_1 , (10-8) n ~ 2 , &c, multiplying each digit by its coeffi- 

 cient and adding the terms thus produced*. The second is a 

 repetition of this process such a number of times as will produce 

 a single digit. 



This method of obtaining the remainder to 8 is called unita- 

 tion. The remainder corresponding to a given number for a cer- 

 tain value of 8 is said to be the unitate of the number, and the 

 divisor (8) the base of the system of unitates under considera- 

 tion. The symbol U$# is used to signify the unitate of the num- 

 ber x to the base 8. 



The principles brought to bear, and the examples given in the 

 former papers upon this arithmetical process, showed that it was 

 useful to check calculations, to obtain remainders to a given di- 

 visor without the use of any multiple of that divisor, and to verify 

 tables. It was further shown that negative and positive integers 

 could be found, by the process of unitation, to represent uni- 

 tates that could not be obtained by division. 



The nature of the operation of unitation, and its position 

 amongst other operations, will first receive attention, as introduc- 

 tory to the determination of the unitates of powers and roots and 

 to the discussion of some of their properties. 



Of the two kinds of operation (necessary for the complete 

 unitation of a number) mentioned above, the first part (involving 

 the addition of terms) is analogous to the formation of an ordi- 

 nary number, in the decimal system say, another multiplier 

 besides 10 being used ; the second operation is analogous to 

 that indicated by the symbol A n in the Calculus of Finite Dif- 

 ferences, or to that of derivation or successive differentiation 

 indicated by the symbol (j> (n) . Unitates, however, do not bear 



* Communicated by the Author. 



t Phil. Mag. S. 4. vol. xxxvi. p. 346, and vol. xlvi. p. 36; and British 

 Association Report for 18/0, Transactions of the Sections, p. 16. 



