Mr. W. H. "W alenn on Division-Remainders in Arithmetic. 351 

 that may be carried from the previous product. In this way 

 the decimal equivalent for ^ is found to be 



0-03448,27586,20689,65517 ; 24137,93i, 



17 1 7 



3 being the multiplier. Since -=—,— = — — -, &c, this 



plan can be used to give the figures of reciprocals that have 7 

 for the right-hand figure of their period : the operation in this 

 Case is begun with 7 instead of 1. By similar ineans, other 

 decimal forms can be constructed ; the forms that can be deci- 



mally wrought out by this means are — ^, —L^ j^, 



and -r— r. The other decimal forms of reciprocals are either 



10a + 1 l 



finite, or they may be derived from the form ^ x-« 



; * J 10a + y 



A method, which is sometimes easier than the above, is to 

 use the multiplier as a divisor, beginning with the last figure 

 of the period, working towards ihe right hand, and taking- 

 each figure of the quotient as it is obtained as the figure to be 

 brought down for the unit figure of the dividend at each ope- 

 ration. The work to obtain = ( = j^ ) by this method is : — 5 



into 7, 1 time and 2 over ; 5 into 21, 4 times and 1 over ; 5 

 into 14, 2 times and 4 over ; 5 into 42, 8 times and 2 over ; 

 5 into 28, 5 times and 3 over; 5 into 35, 7 times and over: 

 5 into 07, 1 time ; and so on, giving the digits (7) -1428571, 

 &c. 



In reference to unknown quantities, the power of imitation 

 to ascertain the remainders to divisors thereof may be shown 

 by an example. The unknown quantity must be connected 

 with a known quantity by definite laws. If, for instance, it 

 be required to take the 647th power of 256 and to ascertain 

 the remainder, to the divisor 7, of that high number, the fol- 

 lowing is the work, taking into account that the function Uj* 

 has 6 for its repeating period : — U 7 256 = 4, and U 6 647 = 5 ; 

 then U 7 4 5 = 2. Therefore U 7 256 647 = 2. 



The function U 5 /, / being a figurate number, has, to some 

 extent, been investigated. U 9 / does not repeat, according to 

 any law that has yet been discovered; but groups of these 

 imitates repeat ; those that have been examined repeat either 

 after 9 terms or after 9n terms. Un/ being formed by a 

 square arrangement of the numbers as they are obtained, re- 

 peats after 11 terms in the horizontal as well as in the vertical 



