Mr. W. H. Walenn o?i Unitation. 377 



application of the theory of functions shows that each unitate, 

 of the normal kind, must of necessity be an integer ; the in- 

 terpretation of symbols points out that the symbol U^^c is a 

 perfect representation of " the unitate of the number x to the 

 base S," and that therefore Uj" x is the representation of the 

 phrase, " any one of the series of numbers \Yhose unitate, to 

 the base S, is ct*." 



27. Many advantages are gained by the fixedness of whole 

 numbers, as in unitation, over the variability of the denomina- 

 tion and meaning of remainders to division. The first instance 

 is treated of in the Philosophical Magazine for July 1873, 



p. 40. By no kind of division could ^, or * 142857, have been 



divided by 9 so as to give 4 as a remainder ; yet, regarded as 



a number which has the same unitate to the base 9 as = — -r =TT>t 



7 + 9 lb 



the result is easily arrived at. Other instances of this occur, 

 especially when S, the base of the system of unitation, is a 

 prime number ; for in U^d'"'^, when S is a prime number, the 

 only fractional unitate is when x is some multiple of S. Con- 

 sequently the intractable recurring decimals -5( = -3), -( = *16) 



o b 



and ^( = -1), in the series of unitates represented by UhA^"', 

 are respectively symbolized and finited as follows : — 

 Ung=Un4=UiiC04) = 4, 



Un|=Un^=Uu(-05) = 5. 



Another instance of simple results being substituted for inter- 

 minable or incommensurable ones is alluded to in the Philo- 

 sophical Magazine for August 1875, pp. 119, 120, in which 

 1711^3 = 5, and Uiix/5 = 4. The above instances all refer to 

 the simplification which occurs in the numeration of unitates, 

 by adopting the definition for them in its rigid integrity and 

 fearlessly working out the results. 



28. The ordinary formula under which numbers are placed, 

 when their properties are to be investigated by algebraical 

 means, may be stated as 



