118 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Let 
§ 4. The Curves of Coefficients. 
Curves of Coefficients. 
K = Po U 0 + Pl( U l + U -l) + 'P2( U 2+ U -2)+ ■ • • 
be a graduation formula ; by plotting the coefficients p s as ordinates against 
their suffixes s as abscissae, we obtain a set of points which, when joined so 
as to form a smooth curve, may be called the curve of coefficients . It has 
been remarked by G. J. Lidstone * that for the best graduation formulae of 
the summation type the curve of coefficients 
resembles the curve of coefficients for 
Spencer’s formula, which is the fifth curve 
in the annexed diagram (fig. 1). Now, in 
the case of the least -square graduation 
formulae, we readily see that coefficients of 
the k = 1, 2, 3, and 4 formulae have respec- 
tively their second, fourth, sixth, and eighth 
differences constant : which shows us that 
the curves of coefficients in the case of 
1c = 1, 2, 3, and 4 are parabolae of the second, 
fourth, sixth, and eighth order respectively. 
The equations to these curves for m = 10 
are, in fact, 
3059 y = 329 - 5x 2 when Jc= 1, 
2600152/ = 44003 
x 2 + ~x i when k = 2, 
4 4 
3343052/ = 77821 - 
1190357 
“ 180““ 
z 2 + 
143 
180 
24640 . 
180 * 
x 6 when k — 3, 
3231 61 5y = 966073 - 79321660 ^ + 299 49] j a- 
576 
576 
41250 . 187 8 , , , 
-W* + 576* when * = 4 ’ 
where x has values ranging from — 10 to 
+ 10 . 
These curves are also represented in the 
diagram (fig. 1). It is evident that the curves of coefficients for least- 
square formulce do not resemble the curves of coefficients for the sum- 
mation formulae, the differences between the curves being so fundamental 
as to indicate an essential difference between the two classes of formulce. 
* Journ. Inst. Act., 41, p. 348. See also Karnp (J.), Trans, of 2nd International 
Actuarial Congress , p. 47, and translation, p. 93. 
