232 
On Corrections for Moment-Coefficients 
blocks such as are indicated in the figure above, reserving for the second pai't the 
treatment of the corrections needful when the frequency curve asymptotes to the 
frequency axis, i.e. the cases of J- and U-shaped frequency distributions. The 
general treatment of non-asymptotic frequency blocks will follow the lines of 
pp. 282-8 of the paper : " On the systematic Fitting of Curves," contributed in 1902 
by one of the present writers to the first volume of this Journal. 
(2) The method there adopted started from the Euler-Maclaurin formula : 
z'dx = {-kz„' + z,' + z.; + ... + z.,_, + \z;) h - h 
, dZ' cPZ' 
1^ "rf^ 720 'dx" 
d?Z^ h7_ d'Z' d:'Z^ _ JiPBn d''Z' 
30240 "d^ 1209600 t^a,'' 47900160 daf 121 dx'' 
where Z' is any function of and Z^', Z/, Z.,' , ...Zp are the ^-1-1 values of this 
function corresponding to i} subranges taken from x = a-',, io x = Xp of the range I. 
Gle-Arly ph = I = Xp — x,). Bi^,By.... are the higher Bernoulli numbers. The first 
term on the right involving the -f- 1 values of the function Z' is the " chordal 
area"; the term between square brackets depends on the values of certain differential 
coefficients at the ends of the range, and these again depend on the form we assume 
for the frequency curve in the neighbourhood of the terminals. The value we are 
going to take for Z' is x^Z, where Z is the integral ydx, or, y being the frequency 
ordinate, Z is the total frequency on the section Xp — a; of the range. In evaluating 
the limits we need not proceed beyond the ninth differential, for the 11th vanishes 
for « = 5 with our assumptions for Z, and in our experience of actual frequency the 
ninth term as a rule contributes very little to the total correction. In order to 
obtain our results we must assume some form for Z at the terminals of the fre- 
quency block. Clearly at x = x^, Z= N, i.e. the total frequency under consideration ; 
at x = Xp, Z = 0. We shall assume Z given by high order parabolae in the neigh- 
bourhood of the terminals, i.e. 
^ _ zv (^1 + , 2 ! V 3 ! //„^ ^ 4l h,' ^ 5 ! V / 
in the neighbourhood of x = Xo, and 
bj^ {Xp - ;/•) 6, {Xp - wf 63 {xp - x Y 64 {Xp - xY b, (xp - xf ) 
h„ 2! /C 3! hj^'^^l 51 h./ J 
Vp ^ . ,Up ^ . lip I . ,ip 
in the neighbourhood of x = Xp. 
These lead at once to 
^)^_^ =iV^a,/V and g^f)^^^ =N{-lYbslh/ (11). 
Exactly as in the earlier memoir we shall determine the a's and b's from five 
frequencies adjacent to the terminals of the range. In many cases, however, 
