548 Mr. W. H. Walenn on Unitation. 



That is, U 10 >' (r being the remainders obtained by subtracting 

 the products, in the usual way of division, from the remainders 

 previously obtained) is always the same as the figure in the 

 quotient which belongs to the previous product. This is a 

 consequence of U 10 9n being the series 9, 8, 7, 6, 5,4, 3, 2, 1- 

 that is, complements to 10 of their multipliers n. 



21. According to another way of regarding like operations 

 to that set forth in art. 20, to have 1 remainder, 39 must have 

 been subtracted from 40, giving 1 for the last figure in the 

 quotient; and the 4 requires 6 to be the unitatc (to the base 10) 

 of the product that gives the next figure, to the left hand, in 

 the quotient. The only product of 39 that gives that imitate is 

 156 ( = 39.4), giving 4 for the penultimate figure in the quo- 

 tient, and so on towards the left hand for the other figures in 

 the quotient by simple inspection of the function U 10 39ft (n 

 being the digital multipliers of 39) ; thus division is performed, 

 by means of unitation, in the opposite direction to that of the 

 ordinary operation. 



22. Reciprocals of the form yrc ^ have 7 for the last figure 



of their period ; for the function ^ — ~ = 7. TT (the symbol 



U io 7 U io y . 



V$\v being read " the number whose unitate to the base 8 is 



equal to a?") ; and (art. 18) TT _ 1C) has 1 for the terminal figure 



of its period. The multipliers to construct these reciprocals 

 from right to left are comprehended in the arithmetical series 

 5, 12, 19, 26, &c., the corresponding denominators being 7 

 17, 27, 37, &c. That this is the series is evident (art. 19) from 



the equations, = = 7. jg. . . having multiplier 4 + 1 , ^ = 7-tjq 



having multiplier 11 + 1, 97 = 7- Ton • • • having multiplier 



18 + 1, and so on. 



Taking U 10 ?' to mean the same as put forward in reference 



to reciprocals of the form =^ ,- (art. 20), but having relation 



only to reciprocals of the form ^ ~, U 10 ^ may be made to 



J. (J 71 — O 



yield the figures of the quotient in the case ^7: ^ by multi- 



Wn — o 



plication by 7 ; this is a manifest result from the equation 



~iH ^ 7 ==7, ^F ^ 9• 



