K. Pearson 
287 
Hence finally reintroducing h for base unit, and substituting in (xi), we have, if 
= V2 
720 "^30240 
«5 - 
\ 
12 120 ' 3024 ' 
/i- A'' /i* 
^^' = ^^'-4^^' + 504"''~960b 
240 126 ' 1440 
04 + 
(xiv). 
5 7 A'" 
5 7 31 A,^ A^" 
= - ^ A^.; + ~ av; - ^-^^ a« - go + 528 " / 
These are the formulae for finding the first six moments about one end of the 
range when we have found the f's or the " rough moments," i.e. the frequencies 
grouped at the mid-points of the elements, about the same end. Putting 
a/ = ag' = ttj' = O5' = . . . = 0, 
we have the corrections first given by Mr W. F. Sheppard {Proceedings of the 
London Mathematical Society, Vol. xxix. p. 368) for the case of high contact at 
both ends of the range. 
It remains now to be considered how we can determine good values for the as. 
I assume that the form of the integral curve near the origin can be closely given 
by a parabola of the 5th order. This, since Z — N and dZjdx = 0 for « = 0, must be 
whence we have as required (^^) =(tis'^=(^s- 
Now when x= e, w = 2€, x = Se, x = 4e. let 
Z=N{l-n,), N(l-n,-iu), N'il-ii.-Ji.-n,), ^(1 - n,- n.- n,- n,). 
Thus we find : 
a^'e'^ a^'e^ a^e* aJe^ 
-^^^ = 12-+-^+ f +15- 
-0h + -) = ^2^ + '^f^^23^^2^ + «^2a 
-(n. + .. + n.)==^^3^ + ^j^33 + «^^3^ + «^3^ 
{f. + + + «.) = ^ 4^ + ^ 43 + ^ 4^ + 4^, 
