120 Mr. W. H. Walenn on Developments and Applications 

 Unitates of Powers and Roots to base 11. 



a 8 . 



i 



i 



a\ 



a 2 . 



« 4 . 



a 8 . 



a 16 . 



>. 



o* 1 . 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



28 



st 



2- 



2 



4 



5 



3 



9 



4 



5 



9 



4 



5 



3 



9 



4 



5 



3 



9 



4 



5 



3 



9 



4 



5 



3 



9 



4 



5 



3 



3 

 i 

 6 8 



7 i 



9 



i 

 6* 



i 



7 4 



4 

 7* 



5 



6 



7 



3 

 3 

 5 



9 

 9 

 3 



4 

 4 



9 



5 



5 

 4 



3 

 3 

 5 



9 

 9 

 3 



88 



8* 



8* 



8 



9 



4 



5 



3 



9 



4 



4 



5 



3 



9 



4 



5 



3 



9 



4 



5 



108 



10* 



10* 



10 



1 



1 



1 



1 



1 



1 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



7. Unitation squares, however, will not always give all the 

 whole-number unitates, on account of the law of the series being 

 sometimes discontinuous, from abnormal causes. In the func- 

 tion U u a n , for instance, the values of the natural numbers in 

 order being allotted to n, the repeating period is found to be 8 

 in the case of U 17 2 w ; this circumstance throws out the law of 

 continuity in the series (to the base 17) 2 1 , 2 2 , 2 4 , 2 8 , &c, so 

 that the 8th power and all higher terms have the unitate 1 ; the 

 series of the unitates of natural powers are 2, 4, 8, 16, 15, 

 13, 9, 1, &c, and of the unitates of powers with common ratio 

 2 are 2, 4, 16, 1, 1, 1, &c. Nevertheless, on inspecting the series 

 of squares belonging to the base 17 (namely 1, 4, 9, 16, 8, 2, 

 15, 13, 13, 15, 2, 8, 16, 9, 4, 1, 17), 2 is found to be the uni- 

 tate of 6~, therefore 11)7^2 = 6 01' 11; this result might have 

 been expected from the fact that 34(=17x2) is (in example I. 

 of art. 3 in the present paper) found to bs one of the values of 8 

 in which i/2 has an integral unitate, 



8. The checking of calculations in which surds occur is there- 

 fore possible by unitation. To know which of the two unitates 

 (in which the function V s \/a occurs) is to be used, the follow- 

 ing point, which applies to the bases 11, 9, and 7, may be pro- 

 fitably borne in mind. The only part of any formula in which 

 a surd occurs, which can affect the unitate of the surd, is the 

 factor by which it is multiplied; the presence of other additive 

 or subtractive terms cannot affect the unitate in any way. That 

 is, the form of a surd value being written a + bs/c, a need not be 

 noticed in reference to this question, and b is the sole quantity 

 that influences the value of Us \/c. When U 5 Z» is odd, the least 



