Mr. W. H. Walenn on Negative and Fractional Unitates. 39 



equal to 



25 + 9 34 



or T' or 5' 



Therefore unitates of all quotients 



may be obtained by preserving them in the fractional form which 

 they originally have — neither neglecting the remainder, nor re- 

 ducing any fraction which may thence arise to lower terms, but 

 preserving the exact unitate of the divisor to the very end of the 

 process, either in reducing the unitate of the fraction which ex- 

 presses the division to be checked, or in reducing the unitate of 

 the quotient (or answer), including the unitate of the remainder. 

 If there be no remainder, the unitated remainder must be a frac- 

 tion which has 9 for its numerator. Division of unitates, there- 

 fore, renders the investigation of fractional unitates necessary. 



Since all divisions can be written under the form a . j, and 



since -=b~ 1 , it follows that the question of fractional unitates 



is identical with that of the unitates of the ( — l)th power of 

 numbers, and also with that of the unitates of reciprocals 

 decimally expressed. In the annexed Table of unitates of 

 the powers of numbers, it is observable that the unitates of 



Uni. 



Uni. 



Uni. 



Uni. 

 a 2 . 



Uni. 



Uni. 



Uni. 

 a 5 . 



Uni. 



Uni. 

 dJ. 



Uni. 

 a 8 . 



Uni. 



Uni. 



a w . 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



5 



1 



2 



4 



8 



7 



5 



1 



2 



4 



8 



7 



3 



1 



3 



9 



9 



9 



9 



9 



9 



9 



9 



9 



7 



1 



4 



7 



1 



4 



7 



~l" 



4 



7 



1 



4 



2 



1 



5 



7 



8 



4 



2 



1 



5 



7 



8 



4 



i 



1 



6 



9 



9 



9 



9 



9 



9 



9 



9 



9 



4 



1 



7 



4 



1 



7 



4 



1 



7 



4 



1 



7 



8 



1 



8 



1 



8 



1 



8 



1 



8 



1 



8 



1 



i 



9 



1 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



the same power of different numbers (the vertical columns) 

 repeat themselves after every nine consecutive numbers; also 

 that (with some exceptions presently to be noticed) the unitates 

 of different powers of the same number (the horizontal columns) 



