1919-20.] On a Class of Graduation Formulae. 
113 
may be called the “ summation ” type, which is based on the following 
principle. Let A denote the operation of differencing, so that Au n — 
Un+i—Un, and let [2m +1] u n denote the sum of (2m + 1) u’s of which u n 
is the middle one, so that 
[2m + 1 ]m 0 = u_ m + u__ m+1 + . . . +u_ 1 + u 0 + u 1 + . . . + u m _ 1 + u m . 
Then it is possible to find combinations of these operations A and [ ], 
which, when differences above a certain order are neglected, merely 
reproduce the functions operated on ; so that we have (say) 
/{A, [ ] } u x = u x -J- high differences. 
We now take /{A, [ Yf u x to be the graduated value of u x ; that is, 
u x = /{ A) [ ~\) u x • 
the merit of this depending on the circumstance, that /{A, [ ]} u x 
involves a large number of the observed u’s, whose errors to a considerable 
extent neutralise each other, and so produce a smoothed value u x in place 
of u x . 
Perhaps the best of the summation formulae of graduation correct to 
third differences is that of J. Spencer,* namely, 
= W . HId (1 — 43 2 — 33* - J3> 0 , 
O.O./ 
where 0% o stands for ^_ 1 — 2 u 0 + %. Written in full, Spencer’s formula is 
u' — o {60 ?£ o + 57(mj + m_i) + 17 (u 2 + w_ 2 ) + 33 (u B + w_ 3 ) + 18 (m 4 + w_ 4 ) + 6(u 5 + u~ 5 ) 
- 2 (u 3 + u_ 6 ) - 5 (u 7 + u_ 7 ) - 5(u 8 + w_ 8 ) - 3 (u 9 + u_ 9 ) - (u 10 + w_ 10 )}. 
It is a case of a more general formula, which may be written 
u ' [p]MM - • • 
0 p.q.r. . . 
| 1 _ 02 + any f multiples of higher even central differences | u 0 , 
where p, q, r are any whole numbers, of which an even number (or zero) 
must be even, and any number may be odd. 
§ 3. Formulae Based on the Method of Least-Squares. 
The formulae of graduation which are studied in the present paper are 
obtained by a wholly different method from the summation formulae 
which have just been referred to. Supposing the ungraduated values^ 
to be plotted as points against the corresponding values of x, we shall fit 
a parabolic curve of some assigned degree j to successive sets of (2m + 1) 
points (u _ m , w_ TO+1 . . . u 0 . . . u m ), determining the constants of the curve 
* Journ. Inst. Act., 38, p. 334. t Usually small multiples. 
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VOL. XL. 
