310 
Miscellanea 
It will be noted at once that the assumptions made are (i) that Taylor's Theorem may be 
applied to the frequency function throughout the range, i.e. {x) and its derivatives must be 
finite and continuous throughout the range; further (ii) that x't^^^x), where s and p are any 
integers, and its derivatives are finite and continuous throughout the i-ange, and (iii) that <p (x) 
and its differentials vanish at the limits. 
Now (ii) practically flows from (i), but (iii) is a weak spot and must be borne carefully in 
mind when the corrections are applied. It amounts to saying that the contact is of an 
indefinitely high order at the ends of the range. This is by no means generally true, the 
curve frequently meeting the character axis at a finite angle, or even being perpendicular to it. 
In such cases special processes must be adopted to correct the moments*. The real trouble 
of the analysis often lies in the fact that until the curve has been fully calculated out, we are not 
in a position to determine the nature of the contact at the terminals, i.e. we want to know the 
moments before we can determine whether they ought to be coi'rected by Sheppard or not. In 
some cases the graphical representation may suffice to indicate the nature of the contact, but it 
is by no means conclusive, often indeed misleading. 
This arises to some extent from the fact that we usually deal with areas, not ordinates, in 
plotting such graphical representations. If we accept an expansion up to /i* as giving approxi- 
mately enough for jiractice the value of n,. in terms of y,,, we can deduce a number of quite useful 
results fi'om equations like (i). For example, let us take the normal curve : 
A' -IK 
Here (,r)=j/ — , 
0'^ (■')=y • 
Hence ...^z^^,,, |i + _ -^,-+^ 1. 
Xow the second and third terms will be largest when .r,. is biggest, say .r,. = 3o- in practice. In 
this case 
But h will hardly be as large as icr, or say about 12 groups. Then we have 
''■=^'-^''{i+r2 + ro2-4}- 
Thus the term in A* even at maximum is for practical purposes negligible, but the term in 'h? 
may amount to as much as 8 per cent. Accordingly we conclude that the frequency may be 
found from the ordinate in the case of the normal curve by the formula 
There is another way of looking at this result. Consider the expression 
N 
and put (T^ = a" -k-pli^, we have on exjianding and retaining powers of h up to A*, 
'''"Va^cr" "I 2<r* + 8" ,x« r 
* See, for example, Biometrilm, Vol. i. pp. 282—88. 
