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



calculations and other ordinary purposes) is that which has 9 for 

 its base. Examples and elucidations of this system, and of its 

 applications to arithmetical work, far beyond what has been 

 realized of it as simply " casting out the nines," have been given 

 in the first two papers on the subject. Its peculiarity is that, 

 all the coefficients being unity, every digit of a given number 

 becomes a datum to obtain the result with the simplest possible 

 arithmetical work, namely addition. By a legitimate extension 

 of the meaning of the word unitate, values of 8 such as 99, 999, 

 &c. come under the same category as 8 = 9 ; for instance, the 

 addition of alternate digits gives the unitate of a number to the 

 base 99, and so on. 



Taking 2, 4, 5, 8, and 10 as values of 8, they would seem to 

 be useless and without any application whatever, since they only 

 include a limited number of terms, and do not necessarily involve 

 all the figures of the given number : this is not the case, how- 

 ever ; for 10 (and its congeners 100, 1000, &c), for example, pro- 

 mise to become most useful for certain purposes. These are the 

 only systems of unitation which give digits in the number itself; 

 and they furnish the means of verifying the last figure, or figures, 

 that the table of logarithms used falls short of, and for other 

 purposes that will hereinafter be more particularly described. 



In a theoretical point of view, perhaps the most perfect of any 

 system of unitates (when 8 is less than 10) is that having the 

 base 7 ; this system has the greatest number of whole number 

 unitates that it is possible for any base to have. The only other 

 system which will be particularly noticed here is that with the 

 base 11. The unitates to this base are obtained with nearly equal 

 facility to those with the base 9 ; they are useful as an additional 

 check to calculations ; and they possess some properties in com- 

 mon with the function U 7 #. 



By taking the ordinary form of the multiplication table, as 

 sometimes put forward in books, so as to form a square*, as a 

 model upon which to construct a table of the unitates of powers, 

 the advantage is presented of having two series of numbers, one 

 at right angles to the other, open to inspection at a glance, one 

 of the series (the vertical lines) recurring in a number of terms 

 equal to the value of 8, the horizontal series recurring in a num- 

 ber of terms dependent upon the value of 8. The following are 

 three of these arrangements (which in respect to the unitates of 

 powers, or to powers at all, are believed to be new), showing the 

 form of each series of unitates of powers, and at the same time 

 that of the series of unitates of powers to the same number f: — 



* See article " Arithmetic " in Encyclopedia Metropolitana, by the Rev. 

 George Peacock, D.D., p. 486. 



t The square showing the function V Q a n was given in the paper upon 

 Negative and Fractional Unitates. 



