378 Mr. W. H. Walenn on Unitation, 



In this formula r is the radix of notation in which the number 

 is expressed, a^, a^, 0.2, a^, &c. are the digits that compose the 

 number, each being less than r; and there are n digits. 

 When the radix is 10, this formula becomes 



a^ + lOcii + 102^2 + lO^as + . . . + lO^'-^an-^ + 10«-^a„_i. 



The mathematical mind has been so much imbued with this 

 formula, or with formulae that are exactly its counterparts, 

 that any deviation from the recognized method of working 

 demands the ofivino- of such substantial reasons as shall assure 

 the reader of the advisability of the alteration. 



It is conceded by all, that the form or method of arrange- 

 ment of a formula must be that which is best suited to the 

 object in view ; for instance, in the theory of equations the 

 general formula is always arranged in descending powers of 

 X, the highest power being towards the left hand. The very 

 life of unitation depends upon the treatment of it in a perfectly 

 appropriate and orderly manner, taking into account analogy 

 and exactness of form to express the general idea that is in- 

 tended. Every number, regarded as a function of its radix, 

 must have its highest power to the left hand. 



Again, in the ordinary formula, nothing Is gained by having 

 the series of digits marked by a^ at the beginning ; on the 

 contrary, by this arrangement the general term does not ex- 

 press the number of digits readily. On these grounds the 

 formula is taken here as 



anr'^~ ^ + a«- 1 r''~'^ + . . . + a^r'^ + agr + a^ ; 



or if r = 10, 



W-'an+10''-''an-i+ ... + Was + 10^2 + %. 



Thus arranged, the formula is a picture of the number in alge- 

 braical language ; this will be seen to be an important point 

 in the method of working with the formula for the purposes 

 of research and investigation. 



74 Brecknock Road, N., 

 October 5, 1877. 



Addendum. — Since writing the above, the Report of the 

 British Association for 1876 has come into the hands of the 

 author. The statement in the Transactions of the Sections is 

 as follows: — ''The author referred to a series of papers of his 

 on unitation recently published in the ' Philosophical Maga- 

 zine,' and to some remarks published in the British- Association 

 volume for 1870. If a; divided by 8 leave remainder?/, th(^n the 

 author calls y the unitate of .^' to the base S, and writes 'Usx=i/. 



