Eleanor Pairman and Karl Pearson 
239 
should have 6/ = ^s, '-(■s' = <^f< and for working units where h generally = 1, Xp =Xp, 
x„' = ,7„. We find 
Ml' = I'/ + h {J^ {a,' - ji^a,' + ^^a^) + (6/ - ^^6/ + ^Ao (xvi). 
+ hx:(a,' - J^a/ + ^TTfln') + i< (V - J^TjW + W?u^/)1 (^vii). 
+ K"(V-oV&3'+^5W01 (Xviii). 
= '^4' - ^rv^ + /i' + {j^^ (do - ^^a/) - jij. {b.: - ^^W) 
+ (xix). 
- (''i' - + ijio^/) - Tf - + tV'''7'" (^2' - rivM) 
+ A^o'^ («/ - + + (^1' - tAt^/ + -5^0^^')} (XX). 
- AV' (^2' - ^^4') - («/ - (/!j«3' + ^ji^'r!) - I x;^ iW - + ^hW) 
- h^o' («'/ - jh^t'i) + s^p' (^2 - ih^^') + ("1' - uV"-/ + TjgL-a/) 
+ h^-p' iW - + ^^jjh^)] (xxi). 
The first series of terms outside the curled brackets are precisely the Sheppard's 
corrections for the moments which accordingly still remain essential portions of the 
corrective terms even when there are final terminal ordinates and any degree of 
abruptness in the slopes at the end of the range. We may speak of the a's and b's 
as the "abruptness coefficients." They are determined by equations (iii) and (iv). 
It will be clear that the terms in the curled brackets repeat themselves, so that 
in working as we usually do to the fourth moment we have to deal only with eight 
functions of the abruptness coefficients. 
The next stage is to consider how equations (xvi) — (xxi) may be most ad- 
vantageously arranged for practical statistical work. In all such work the subrange 
h is taken as unity. Hence we may always write it 1. Further, the origin is at 
our choice, and it might seem desirable to take it at the mean. But there are two 
means, namely the true mean /u./ of the data and the mean v/ of the concentrated 
groups; with abruptness these are no longer identical. If we take moments about 
the true mean, is not zero and our calculations are not simplified by its vanishing. 
On the other hand if we take moments about the mean of the concentrated groujis 
