Miscellanea 
383 
which is formed from equations II. and V. in order to find the terms to be used for giving 
the corrected moments. The exact area may not be reproduced, for if we find the first term 
by XV. and the other terms by XIV. and then use XVI. we have the following result : 
I " y^dx = 1 {26^0 + 21i/i +25^0 + My, + . . .} 
= X J-{26x23Jo + 26x2J,-26^2 
24 24 
-21^1o + 21 x26 ^11-21^2 
-25yli + 25 x 26.42-25^3 
— etc.} 
= ^{577ylo + 573/li + 579.l2 + 575.l3 + 576il4+...}, 
576 
or the area is overstated by 
^{.io-3.1, + 3.l2--l3}. 
When a pai-abolic curve is being fitted these formulae can be applied as they stand, for 
in such cases we can generally choose our range to some extent, that is the formula itself does 
not fix an actual starting place for our curve ; but in much frequency curve work the difficulty 
of adjusting the moments is, as has been already remarked, considerably increased owing to 
the rapid rise of the curve from zero. In such cases, especially when the terms are few in 
number or there seems a likelihood from the original statistics that the curve does not start at 
the beginning of the base of the first group, I think the formulae should be applied neglecting 
the first group, which should be examined separately. . 
Taking the following un^jromising series (col. 1) from Phil. Trans. Vol. 197 A, pp. 454 — 456, 
the following table was made. 
TABLE VII. 
Frequency 
(1) 
Ordinate by 
XIV. & XV. 
Modified ordinate 
by XVI. 
47 
(neglected) 
(neglected) 
762 
742-7 
804-6 
160-5 
141-2 
123-6 
20 
14-8 
15-4 
5 
4-5 
4-5 
1-5 
1-3 
1-3 
996 
If the curve starts at the middle of the base of the 47 group its mid-distance from the middle 
3 
point of the base of the 762 group is really - and this should be taken into account in 
calculating the moments. Now it is easy to see that the first group probably relates to a small 
base because the mid-ordinate calculated from ^ (23/lu + 24i - J2) is —8. If in order to 
