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



repeat themselves after every six consecutive powers. In this 



Uni. 

 way the column for reciprocals, headed « -1 is obtained, so far as 

 the numbers 1, 5, 7, 2, 4, 8 are concerned; for these are also 

 the unitates of the corresponding decimals to the reciprocals, the 

 unitate of 7 _1 or of j being obtained from theunitate of the de- 

 cimal value of 1 1 £ = '0625. To test the correctness of these uni- 

 tates, the principle that every number is the reciprocal of its 

 own reciprocal may be applied ; accordingly the following equa- 

 tions are true in unitates, bearing in mind that, in the case of 

 unitates, the signs +, — , x, -r-, and = have not the same 

 signification as in ordinary arithmetic. The words " the unitate 

 of " are understood to be placed before every numerical value set 

 down. 



2=4 = ^±i 



2. 



3= y = lx3 = 3. 



l_ J3x9) + l _28 



7~ 7 "7 



5 ~2" 2 ~ 2 - 5 ' 



6= ^=1x6 = 6. 



y= l = (3x9) + l = 28 = 7 



4 4 4 



„ 1 (7x9) + l 64 

 8== 8 = 8 = 8" =8 - 



9= *=lx9=9. 



Another property of reciprocals that will test these results is, 

 that the product of any number multiplied by its reciprocal is 

 always equal to unity. This is found to be true in carrying out 



Uni. 

 the column headed a -1 in the above square Table according to 

 this principle; and it is clear that neither 3, 6, nor 9 multiplied by 

 any other whole number can give a product whose unitate is 1. 



The operation of multiplying the unitates by their reciprocals is 



1x1=1. 

 2x5 = 10=1. 



3xi=| = l. 



4x7 = 28 = 10=1. 

 5x2 = 10=1. 



6xi=f=l, 



7x4 = 28 = 10=1, 

 8x8 = 64=10=1, 



9xi=t=l. 



It is to be remarked that the discontinuity of the series in the 

 case of the unitates of the powers of 3, 6, and 9 respectively, 

 which is evident from the second and higher powers not follow- 

 ing the same law of recurrence as the first and powers below the 

 first follow, is an indication of departure from a general rule, 



