240 
On Corrections for Moment-Coefficients 
we shall then have to transfer the fi's to their true mean. Nothing is therefore 
gained practically by the process. Besides this, neither mean is a good working 
origin, and if we take such to calculate the moments in the first place we may have 
two transfers to the means, one for the v's and one for the /x's. Further Xp and oc^ 
will botli have to be calculated and all the terms used. We can, however, get rid 
of slightly less than half the corrective terms, if we take moments about one end of 
the I'ange* and then transfer the /a's to their mean. This appears to us in practice 
to be the best policy, for, although it involves taking the differences of large numbers, 
it is quite easy with modern mechanical calculators to retain the requisite number 
of figures for accurate results. Accordingly we will rewrite our formulae with these 
changes, remembering that is now the range I = ph = p. 
= + (tV («/ - + W^^O + tV (V - jkW + (xxii). 
= "/ - tV + 1- ih - tIg «/) + i^- - ih^^) 
+ ip(P^-lhW + ^^.oW)] (xxiii). 
= v,'- } v; + {- (a/ - ^a/ + - (h^ _ + ^^^h^;) 
+ -hp iW - jhW) + Ip"- iW - ^^h,' + ^^jjh')] ' (xxiv). 
f^: = - i V,' + + _ _ _i_ _ iZ,;) - -^-p (6/ _ J'^ b,' + ^\^W) 
+ iofil>^-ThW) + ^f(h/-,hW + ^rkoh;)} (XXV). 
- ihp (w - ik^:) - \p' (w - j^b/ + ^yw) + i^p' ih.; - j^b:) 
+ i^P'{b; -h^; + ^^T>b;)] (xxvi). 
+ ikP (V - 4^^.' + ihW) - ^kP" - hK) - \f {K - jhbs + ^K) 
+ W (^2 tIo + hf - tjV^/ + ^hoW)} (xxvii). 
(6) We now propose to illustrate the degree of exactness with which it is 
possible to obtain the moment-coetticients of curves with marked degrees of abrupt- 
ness, and further to investigate in pi-actice the extent to which small terminal range 
elements may be of advantage. We will commence with some mathematical 
frequency distributions for which it is possible to calculate the exact values of the 
moment-coefficients. 
lUustratiun I. Moment-coefficients of the common parabohx ij = V,r x 100,000 
from X = 0 to 10. This is a good case for a test, for the curve rises vertically at 
^0 = 0, and therefore, theoretically, our equations fail. At x,, = 10, we have a finite 
ordinate and finite abruptness coefficients. We are hardly likely to get a case 
wherein the abruptness causes greater changes in the grouped frequency moments, 
or to which it is less possible a priori to apply merely Sheppard's corrections. 
* The distances of the successive concentrated groups are ^h, ?jh, %h, ... . In taking rooments it is 
convenient to use 1, 3, 5, etc. and then before substitution multiply by ('5Y. 
