then 



218 Mr. W. H. Walenn on Unitation. 



III. If the formula with the reduced coefficients be em- 

 ployed*, namely 



U 7 N = a 7 + 5a 6 + 4« 5 + 6a 4 + 2a 3 + Sa 2 + a ly 



U/ 1^=1 + 10 + 12 + 24 + 10 + 18 + 7 = 82. 



U 7 "N = 3. 8 + 2 = 26. 



U/ // N=3.2 + 6 = 12. 



U 1 / N = 3. 1 + 2 = 5. Here(n) = 4. 



34. In 1ST,, as soon as any value of a u is increased, by the 

 successive addition of units, up to or beyond r, it is trans- 

 ferred to the next higher term, or that containing the factor 

 a n+1 , by adding a unit to the higher term and placing the 



remainder to r. or of the division — , in the term in which the 



lower factor a n occurs ; that is, r determines the maximum 

 value of a n in each term. 



In UiN, on the other hand, 8 may be taken of any integer 

 value in respect to r, and the formula will still be true, but r 

 will have no power to determine the highest value of a n in any 

 term ; & is the only determinator of the maximum value of a n 

 in any term. For illustrations of this see Philosophical 

 Magazine, May 1875, p. 347, and the above instances of U 9 N 

 and U 7 N. 



35. The value of S, whether integral or fractional, for instance, 

 determines the degree and kind of discontinuity that exists in 

 U$N. For example, in IL67 = i, ^ is taken, by inference, as 

 the unit; the same occurs in Uc>i67 = ^. In U2x25 = J=lJ, 

 ^ is the unit. 



36. In regard to the arrangement of the terms, IN" gives 

 simply the arrangement of a number in powers of r ; whereas 

 UsN gives the arrangement of the same number in powers of 

 (r — S). In the most useful form of U 5 N", each power of 

 (r— B) is reduced by substituting for it its remainder to £. 



74 Brecknock Koad, N., 

 December 1877. 



* See Phil. Mag. May 1875, p. 347. 



