114 Proceedings of the Koyal Society of Edinburgh. [Sess. 
by the method of least-squares, and we shall then take the successive 
central ordinates of these curves as the graduated value of u 0 . This 
method has not hitherto been adopted in actuarial practice, and indeed the 
detailed graduation formulae based on it do not appear to have been given 
previously in extenso , except for small values of j, although the theory 
of the method has been discussed, examples given, and many interesting 
properties established in some valuable papers by W. F. Sheppard.* 
Among the questions which present themselves naturally, but have 
not as yet been solved, may be mentioned the following. What is the 
connection between summation formulae and least-square formulae ? and, 
in particular, is it possible for a formula to belong to both types — or, in 
other words, are there any formulae which can be justified by the least- 
square principle, and can moreover be represented by the summatory 
notation ? Such formulae would, if they exist, be of particular interest, 
since the process of summation is so easily performed by the computer 
that summatory formulae are the simplest to use, whereas the least- 
square formulae appear to have the sounder theoretical basis. Again, is 
it a fact that least-square formulae give better graduation than summation 
formulae ? Or are the least-square formulae better adapted for dealing 
with certain classes of problems and the summation formulae better 
adapted for dealing with other classes ? These and similar questions 
provided the motive for the present investigation. 
Let u x be the ungraduated value of u corresponding to the value x, 
and let u\x) be the ordinate of the parabolic curve which is fitted to the 
data, so that the problem is to find 
u(x) = C 0 + CjX + C 2 X 2 + C ? X 3 + . . . + CjX\ 
m 
for which ^ {u P — u\p )} 2 is a minimum. 
p=-m 
The “ equations of condition ” are 
c 0 + c^ni + c 2 m 2 + c s m 3 + . . . 
+ Cjin* 
c 0 + ci(m - l) + c 2 (ra- l) 2 -f c B (m- 1) 3 + . . 
. + Cj(m - 1 y 
— u m — i 
c 0 + c x .l+c 2 . 1 2 + C 3 . I s 
+ Cj . 
= % 
c o 
= u 0 
e o~ c i- 1 +c 2 . V - e s . I 3 
+ (-i)V J ' 
= U- ! 
c 0 - cfm— 1) + c 2 (?7i - l) 2 — c 3 (m— 1) 3 + . . 
c o — G i m + c 2 m2 — c 3 m3 + . . . 
+ ( - 1 )fy» 3 ' 
1 U-m- 
* Proc. of the Vth Internat. Congress of Mathematicians , Cambridge , 1912, ii, p. 348 ; 
Proc. of the London Mathematical Society , ser. 2, vol. xiii, part ii ; Journ. Inst. Act ., 48, 
p. 171 ; 48, p. 390; 49, p. 148. 
