Evolution by Subtraction, 193 



again, this series is the one whose general term 

 ttji =(n»-n)-{(n-l)"-(n-l).}. 



If we substitute 1, 2, 3, <fcc. for n in this formula, we shall ob- 

 tain the successive subtrahends for any given index m. 



No confusion need be apprehended from the double sense 

 in which I am using the symbol ?i, to indicate both the required 

 root and the ordinal number of each term in the series ; for 

 during the operation the value of the root is unknown to us, 

 and we substitute the integers successively in the hope of find- 

 ing it ; and a consideration of the series will show that the 

 number of terms required is in every case = n, including the 

 zero term for n=l. 



The general term as given above is not yet in a form adapted 

 for simple arithmetical calculation, except for very low values 

 both of m and. n ; but by expansion we may cast it into a form 

 capable of resolution into factors in such a way as to make 

 the calculation easier. For 



( n m— ?i)— j(n — l) m — (n — 1)\ =n m — l — (n — l) m ; 



which, expanded by the Binomial Theorem, the highest powers 

 of n cancelling, leaves us 



u n = — l+mn m ~ 1 — ^m(m — l)n w - 2 + &c±mw+l. 

 Here, if m be even, the last term is —1, and the expression for 

 « n =2{ — l + imn(n m - 2 — \{m — l)n m ~ 3 + &c. 



-i(m-l)n + l)\; 



if m be odd, the last term is +1, which cancels, and the ex- 

 pression becomes 



u n = mnQi m - 2 — J(m — l)n m - 3 + &c.-\-^(m — l)n — l). 



In this latter case the quantity within the bracket will always 

 be divisible by (n — 1), as may be seen by an inspection of the 

 coefficients. So far, then, the method appears to favour the 

 odd indices, which might have been expected to present the 

 greatest difficulty ; and an even index, if not a power of 2, 

 may be reduced to an odd one by one or more operations for 

 the square root. 



But an alternative rule offers itself, of making each term of 

 the subtrahend series to be itself the sum of a series. Such a 

 series will of course be that of the differences of successive 

 subtrahends; and in working we shall have to add to each 

 term the last preceding subtrahend, which will have been tho 

 sum of all the previous terms. The general term of this series 

 will be u n+ i—u n . Now this expression 



= (n-\-l) m — 1 — n m — \n m — 1 — (n — l) m | 

 = (n + l)" + (n-l)*-2n* 



Phil Mag, S. 5. Vol 10, No. 61. Sept. 1880. P 



