198 The Rev. F. H. Hummel 



on 



Having now the actual numbers before us, we can verify in 

 fact the relation which we may find from the algebraic for- 

 mula? to exist between the terms of consecutive series. De- 

 noting the number of a series by a bracketed index, we may 

 write this relation thus: tt< w+1 >«nt*f ,) + S£l ) I + 2 ( n "" 1 ) » the 

 nth term of any series =n times the nth term of the preceding 

 series + the sum of the (w— 1) previous terms of that series 

 + twice (?* — 1). Thus, having given five terms of the 6th 

 series, the 5th term of the 7th series will be 



5 x 11528 + 62 + 664 + 3366 + 2 x 4^ 61740. 



By this formula we may extend our tables to any number of 

 series, and to any number of terms in each. 



We may notice in passing the recurrence of final digits of 

 every five terms of each series, repeated in every group of four 

 series, thus : 



m = 4p, 6,4,4,4,8; m = 4p + l, 6,0,0,0,6; 



m=4p + 2, 6,2,4,6,8; m = £p + 3, 6,6,8,6,6. 



So the sum of hp or hp + 1 terms in every series, and of hp ■— 1 

 terms of the odd series, have zero for the final digit,' as would 

 be necessary for powers of numbers of the form 5p or 5p±l. 

 All the terms of each series are even numbers, as we might 

 have foreseen from the algebraic forrnulse. This, too, was ne- 

 cessary ; for the powers of odd numbers being all odd, and 

 those of even numbers all even, n m — n is in every case even. 



So much, then, for the mode of constructing the subtrahend 

 series. Before proceeding further, I may remark that this 

 method is of course as useful for involution as for evolution ; 

 for being in possession of the series whose sum to n terms is 

 7i m —n, we have only to add that sum to n to obtain the value 

 of n m . 



Hitherto we have seemed to assume that the number whose 

 mth root is to be taken is the exact mth power of an integral 

 root. But clearly, if this were an indispensable condition for 

 the applicability of the method, its use would be extremely 

 limited. Let us consider the case in which the given number 

 is not an exact mth power; we may write it n m ±:d. As the 

 evolution is performed throughout by the process of subtrac- 

 tion, every remainder will differ by the same excess or defect 

 ±d from what it would have been if the given number had 

 been n m exactly. Accordingly, our last remainder will be 

 n±d; that is, we have left a number which differs from n by 

 the same excess or defect as that by which the given number 

 differed from n m . As this is by far the most usual case, it is 

 evident that the last remainder is not, after all, the right place 



