Mr. W. H. Walenn on the Unitates of Powers and Roots. 347 



upon themselves any symbol to show the number of times that 

 the operation is repeated to produce the result. This indication 

 might be useful in certain instances; and the results of it have 

 yet to be worked out. 



Comparing the two methods at present known by which the 

 remainders to a given divisor can be obtained, namely the opera- 

 tion of division and that of unitation, it will easily be perceived 

 that obtaining remainders by division is an inverse process, 

 whereas unitation is a direct process. Division has to be worked 

 out by commencing with the highest or left-hand figure, and 

 proceeding towards the smallest or right-hand figure. Unita- 

 tion is begun at the right-hand figure and proceeds towards the 

 left-hand figure. In unitating to the base 10, for instance, the 

 unit figure is identical with the unitate ; for all the powers of 

 (10 — 8), and therefore the coefficients of all the terms containing 

 that factor, become =0. 



These remarks respecting the operation of unitation do not 

 relate in any way to the best and shortest method of obtaining 

 the unitate of a given number to a given base. This was to 

 some extent elucidated in the first article* upon the subject, in 

 which it was shown to be sometimes advantageous to commence 

 at the left hand and reduce the work as soon as possible. For 

 the purposes of investigation, it may frequently be desirable to 

 retain the coefficients in the same form as the expression cited in 

 the commencement of this article furnishes them, and to proceed 

 with the operation as there indicated f. In practice, when ne- 

 cessary, the coefficients themselves may be unitated to the base 8 ; 

 this either produces recurring series as coefficients, or else can- 

 cels them jtogether with the terms to which they belong. 

 The right-hand figure being the coefficient of the unit belonging 

 to the given number, the following are the series for some values 

 of 5 :— 



= 6 ... ., 



• i • 3 



' ) 



4, 



4, 



4, 



1 



= 7 ... ., 



I, 5, 



4, 



6, 



2, 



3, 



1 



= 8 ... 









4, 



3, 



1 



= 9 ... ., 



., 1, 



1, 



1, 



1, 



1, 



1 



= 10 ... 













1 



= 11 ... ., 



• i ' t 



1, 



-1, 



1, 



-1, 



1 



= 12 ... ., 



.,-8, 



4, 



-8, 



4, 



-2, 



1 



= 13 ... ., 



.,-9, 



3, 



-1, 



9, 



-3, 



1 



Unquestionably the most useful and practical of all the sys- 

 tems of unitates included in the above formula (for checking 



* Phil. Mag. S. 4. vol. xxxvi. p. 346. 



t For example, in the function V 7 .v, the above formula or expression 

 becomes 



3"-i a +3 n - 2 b+3 n -»c+ ... +19,6832+6561 m+2\87n+729o+243p 

 + 81g+27r+9s+3*+«. 



