Mr. W. H. Walenn on Division-Remainders in Arithmetic. 349 



to 8, or not a submultiple thereof ; for instance, U 9 ^ = ^. The 

 laws of reciprocals should be satisfied by unitates ; thus 



U n 3 x U n k should be equal to unity, and accordingly 



U 11 (3x4) = U 11 12 = l. 



The function U 5 <£ w , in the case of 8 being a prime number 

 and n being the variable, recurs in 8 — 1 terms. When 8 is 

 not prime, the period of recurrence is less than 8 — 1. When 

 x is the variable and n an even number, it recurs in 8 terms, 

 and 8 — 1 of those terms are arranged in groups, the latter half 

 of each group being the same as the first half but in inverse 

 order ; thus 



IV' 4 =1, 5, 4, 3, 9; 9,3,4,5,1; 11; 1, 5, 4, 3, 9 ; &c. 



The function J] 5 x~ n behaves itself like U^t? -1 in respect of 

 its values being integral or fractional. 



The function U s x n may sometimes be ascertained in whole 



numbers from continuing the series JJ s x mn . When 8 is of the 



i 

 form p n — x (;i and x being whole numbers, and x n or fyx 



\_ 

 being irrational), ~U s x n is finite and integral. In unitation, as 

 in some other numerical operations, there may be m mth roots, 

 n nth roots, and so on. 



Taking the function U 10 »# as true when x = '%/q (an irra- 

 tional quantity), it can be proved to be possible to assign the 

 values of the extreme right-hand figures of some incommen- 

 surable quantities. For instance, 



IW5 = 5, U 100 |/29 = 9, U 100 ^/13 = 17. 



The formula U 5 (— x)~TJ s (8 — x) may be applied to func- 

 tions of V — 1; thus, ~U s (a + b^/ — c) = TJ s (a + b\/8 — c). In 

 this manner *J — 1 in unitates may be expressed as a quantity 

 that is not imaginary, without the use of geometry. 



The investigation of the function JJ s x has not proceeded 

 beyond this point; but several applications of its properties 

 have been made. 



Checking tables and calculations that can be finited is 

 among the most practical applications. All divisions to be 

 tested are obliged to be completely finished by obtaining all 

 the figures of the quotient together with the remainder ; in 

 this case the division of unitates may be performed in the 



