124 Proceedings of the Royal Society of Edinburgh. [Sess. 
comparing columns 9, 10, and 11 : the result is that the sum of the squares 
of the residual errors is 873 when the least-square formula is used, and 
1327 when Spencer’s formula is used. 
y. In Table IV twice the fifth powers of certain numbers are forced, 
and these are the ungraduated values. In this case formulae correct to 
fifth differences must be used. As there is no existing summation formula 
correct to fifth differences which has the same outstanding merit as 
Spencer’s formula has among formulae correct to third differences, two 
fifth-difference formulae, due to G. King ( J.I.A. , 41, pp. 544 and 553), have 
been used along with the least-square formula k== 2, m = 8. These formulae, 
here denoted by (a) and (b) respectively, are as follows : — 
(a) 2187< = [3] 5 { 8 1 [1] + 3[3] - 21[3]2 + 4[3] 3 K 
= 729w 0 + 590^-j- 1 -1-250w ± 2 — 85w ± 4 - 41w ±5 -f- \\u ±1 + iu± 8 . 
(b) 2187V = [3] 6 {84[l]-48[3] + 7[3] 2 }^ 0 
= 7 29z^ 0 + 560^-t-j -f- 280^ ±2 - 70u± 4 — 56u ±5 + 8u± 7 -{-7u± 8 . 
The merits of the graduated values are obtained by comparing columns 
11, 12, 13, and 14: the result is that the sum of the squares of the residual 
errors is 1669 when the least-square formula is used, 2327 when the 
summation formula (a) is used, and 2162 when the summation formula (b) 
is used. 
The general conclusion we have reached may be expressed thus : — the 
least-square formulae of graduation remove accidental errors more 
successfully than the summation formulae of graduation ; but in many 
classes of investigations, the superiority of the least-square formulae is 
perhaps not so marked as to compensate for the somewhat greater amount 
of arithmetical calculation involved, especially when (as in actuarial 
applications) the graduation extends to a series of nearly 100 values. 
§ 7. Sums of Powers. 
Incidentally, in the course of the calculations it was necessary to 
calculate values of the sums of the powers of the natural numbers. As 
the results may be of use to other workers, they are given in Table V 
below. 
In conclusion, I have to thank Professor Whittaker, at whose suggestion 
this investigation was begun, and Mr G. J. Lidstone, F.I.A., for valuable 
help during its progress. 
