8. 



P. 



2 



1 



3 



2 



4 



2 



5 



4 



6 



9 



7 



6 



8 



2 



9 



6 



10 



4 



350 Mr. W. H. Walenn on the Unitates of Powers and Roots. 



The double vertical lines in the above diagrams show the 

 period of recurrence in the unitates of the positive powers of a 

 given number ; this, in the function V 7 a n , is 6 terms, in U l0 a w 5 

 4 terms, and in 1J n a n , 10 terms. The Table in 

 the margin gives the periods of recurrence for the 

 values of 8 up to 10, P being the corresponding pe- 

 riod. In the above " unitation squares/' the series 

 of unitates is continued towards the left hand to 

 show the law of formation of \J s a~ n , which in some 

 cases is different from that of V s a +n . By means 

 of these unitation squares, the unitate of any power 

 of any number to the given base may be calcu- 

 lated by inspection, knowing the recurring figures 

 and the period of recurrence ; for instance, 



U 9 43 562 =U 9 7 4 = 7; U 7 256 647 =U 7 4 5 =2 ) 



u ii 52176345— u ii° - 1 ' 



Practically there is not much difficulty in finding V s ^/q 

 when it is finite or integral ; for it may be ascertained from the 

 root itself (by extraction), or from looking out V$ q in the table 

 of unitates belonging to the series V$ a m and ascertaining the 

 corresponding value of \J s a. It will, however, sometimes be 

 found that there are several values of XJ$ a that will fulfil the 

 conditions; for instance, U 9 y / 729 = U 9 ^/9; and, looking for 

 9 in the series 1, 8, 9, 1, 8, 9, 1, 8, 9, ... , it is seen to belong to 

 3, 6, or 9 * ; a method of ascertaining the true unitate by means 

 of a unitation square will be given further on. It will be dis- 

 covered upon examination that in unitates, as in some other nu- 

 merical results, there may be m mth roots, n nth. roots, and so on. 



Considering unitates as remainders, and inspecting the series 

 of unitates in unitation squares, it will be evident that the unitates 

 of some surd roots have finite and integral values. These values 

 may be determined in the same manner as the unitates of rational 

 roots ; for instance, from the series U 9 a 2 ( = 1, 4, 9, 7, 7, 9, 4, 1, 9, ... ), 

 U 9X /7 = 4 or 5. Further, it will be found, from these reasons, 

 that U 5 %/q (m and q being whole numbers, and tyq being irra- 

 tional) is finite and integral when 8 is of the form n m —q; in 

 this case n is any whole number excepting unity, taking care to 

 choose n so large that the expression n m — q may not be negative. 



Unitation squares, formed in geometrical progression with 

 respect to exponents (in the horizontal series), may be used to 

 obtain the unitates of certain roots, the base of the system being 



* U 11 v / 729 = U 11 ^3 = 9. 



