272 Mr. W. H. Walenn on Unitation. 



5 of the result given by the operation U 5 (n— U 9 n). For in- 

 stance, to obtain U 45 671, 



U 9 671 = 5, 



U 5 (n-U 9 n) = U 6 (671-5)=:U 5 666 = 1, 

 and 



m=— 1 or 4. 

 Therefore 



U 45 671=5 + 9.4=41. 



43. In the set of unitates U 63 N, 



U 7 ( % , a 2i a 3 , . . . aj = 2, 4, 6, 8, 10, 12, 0, 

 and m = | { U 7 (n — U 9 n) } . For instance, to find U 63 1002, 



U 9 n=3, U 7 (n-U 9 n) = U 7 999 = 5, m=i(5 + 7) = 6, 



and U 63 N=3-f9.6=57. 



44. The operations for obtaining m, in the formula 



when k =9, may be tabulated as follows: — 



kn. 



m. 



kn. 



m. 



18 



U 2 (N-U 9 N). 



90 



-U 10 (n-U 9 n). 



27 





99 





36 



U 4 (n-TT 9 n). 



108 





45 



-TJ.Ck-U.K). 



117 



4{U 13 (n-U 9 n)}. 



54 





126 



J{U 14 (n-U 9 n)}, 



63 



*{TT 7 (H-U 9 n)}. 



135 





72 



U 8 (n-U 9 n). 



144 



i{U 16 (N-U 9 N)}. 



81 









The blanks indicate that the operation is impracticable for 

 the corresponding value of kn, as shown in paragraph 42. 



45. It should be observed that, when m is a fractional part 

 of U^(n-U^), as in U 117 n = U 9 n + 9 4{U 13 (n-U 9 n)}, 

 U«(n— UfcN) must be made an exact multiple of the denomi- 

 nator of the fraction by the addition to it of one of the mul- 

 tiples of n. Thus, in finding U 117 25763, 



U 9 . 13 n=5 + 9 4{U 13 25758}=5 + 9.J(U 13 5) 

 =5 + 9. i(5 + 13) = 5 + 9. 2=23. 



46. When £ = 11, the operations for obtaining m take a 



