19-4 The Rev. F. H. Hummel on 



The first term of this addition is cancelled by the — 2a m ; 

 the expression will accordingly be the double of the positive 

 terms of the expansion of (n — l) Wi , beginning at the third 

 term. If m is even, the last term will be + 2. Therefore we 

 may write as an alternative formula, 



?i™- 4 \m , ii m ~ 6 





3.4 \m-Q 3.4.5.6 

 + &c. [ + 2 if in is even] . 

 All these coefficients are integral. When m is even, they will 

 recur in reverse order, like those of the Binomial Theorem, 

 and the whole expression will be divisible by n 2 } with the final 

 2 for remainder. When m is odd, the whole expression will 

 be divisible by n ; and when in is prime, it will also be divi- 

 sible by in. When m— -1 is prime, it will be divisible by 

 (in — l)?i 2 with remainder 2. When m is even but not a mul- 

 tiple of 4, it will be divisible also by (n 2 + 1) with remainder 2. 



We have thus for any integral value of in, whether odd or 

 even, two alternative formulae, in either of which, by substi- 

 tuting 1, 2, 3, &c. successively for n, we may obtain the terms 

 of the required series, either directly, or by summing an an- 

 cillary series ; and by continuously subtracting these terms in 

 their order from the given quantity n m , we should at last find 

 n as a last remainder, at a point beyond which no further sub- 

 traction is possible, since every term of the series is greater 

 than n. As both these formulae depend upon the value of m, 

 it will be possible to establish by means of them an arithme- 

 tical rule of evolution for any positive integral index whatever; 

 it becomes simply a question of ingenuity in each case to cast 

 one or other of the formulae into the most convenient shape for 

 working. 



The rule already given for the square root may be derived 

 directly from the recognition of n 2 — n=n(n— 1) as double the 

 sum to n terms of the series of consecutive integers ; but by 

 our formulae u n+l = 2n ; or ii n+ i—u n =2, the difference of suc- 

 cessive even numbers. So for the cube root, n n +i— w»=6w, 

 or u n+J = 3n(n + l) — formulae which correspond to the two 

 forms of the rule as stated above. In like manner we may 

 find a rule for extracting the 4th root thus : for the (n+l)th 

 subtrahend, multiply ii 2 by 12, and add 2 + the previous sub- 

 trahend ; for if m=4j u n+1 — u n = 12n 2 + 2. 



Now we are well outside the pale of the old rules, which 

 stopped short at the cube root. I propose to venture a few 

 steps into the unexplored region, and find series and rules for 

 some of the higher indices by means of the formulae already 

 established. Taking the integers in order for values of in, and 

 working out their series, we may discover from the lower 



