Mr. W. H. Walenn on Unitation. 



273 



different tone, and the only values of n that present impracti- 

 cable cases are those in which n is an exact multiple of 11. 

 The table of values for m up till n = 16 is as follows : — 



kn or 8. 



m. 



kn or 8. 



TO. 



22 



U 2 (n-U„n) 



110 



U 10 (n-U u n) 



33 



-U 3 (n-U u n) 



121 





44 



-U 4 (n-U u n) 



132 



- U m (n-Uun) 



55 



U 6 (n-U„n) 



143 



-i{U,^H-U u N)} 



66 



-U.(n-U u n) 



154 



-i{U 14 (N-U n N)} 



77 



i{U,(H-U u H)} 



165 



-i{U u (N-U u N)} 



88 



i{U,(H-U u H)} 



176 



-i(Utf(*-U u H)} 



99 



1{U 9 (N-U U N)} 







47. Examples of U n „ N : — 



I. Find U 55 1257. 



U 55 n=3 + 11{U 5 (1257-3)}=3 + 11(U 5 1254) 



= 3 + 11.4=47. 



II. Find U 66 83253. 



U 66 N=5 + ll(-U 6 83248) = 5 + ll.U 6 (-4) = 5 + 11.2 = 27, 



III. Find U 99 75316. 



U 99 N=10 + ll(iU 9 75306) = 10 + ll.U 9 f =10 + 11. 6 = 76. 



IV. Find U 165 6782. 



U 165 N=6 + ll(-iU 15 6776)=6 + ll.U 15 (-.n)' 



= 6 + 11.1 = 17. 



48. When h is a prime number, as in U 7n N, the function 

 behaves itself like U U »N in respect to its general shape and in 

 its comparative freedom from impracticable cases ; at the same 

 time other properties are brought out and a fresh series of 

 divisors are arranged under the formula Ua^n. 



49. The examples of Ui„N are devised to show the prin- 

 ciple involved by the aid of only a few figures. With a little 

 practice, the work may be performed mentally or with only a 

 sparse use of writing-materials. 



520 Caledonian Road, N, 

 ^February 1880. 



