James Henderson 
169 
Then if Cn is the coefficient of 6'^ in <^ {6) and c,/ is the coefficient of O"- in (/>' {6), 
n 
(4) In the last expansion it might seem possible to get rid of the terms in 
Ti and T.2 by breaking away from Laplace and expanding with regard to g-^^V? in- 
stead of e~''^"'P' ; then choose q to give us the desired result. In Laplace's form of 
4 / — 
the modal expansion the exponential term is e \dx'/m«\ where u = \ogy and 
means the value of at the mode. 
Now y 
r(y + i)' 
u = log,, y =p' logg x-x- log, r {p + 1), 
du p J 
cl?u _ p\ 
dx- X' ' 
therefore f -7-;) = — ^ = - -7 • 
VofWMode P' P 
If . , — 7T = <i>{— D) where D= -r- and ^ = , we have to find q, so 
r(/ + l) V27rg ^ 
that either the Tj or T2 term or both will vanish. 
By proceeding as before equation (xvii) becomes 
_ tp'+i) log (i-e)_ 
The term in will vanish if q=p' + 1 which is the square of the standard- 
deviation from the mean, but Ti will still be left. However, it does not seem 
likely that any advantage will be gained by departing from Laplace's form of the 
exponential term. 
Having found the two expansions from the mean and the mode respectively we 
shall now proceed to examine the behaviour of the series by numerical calculation, 
but before doing so we shall endeavour to find a similar series for the Incomplete 
B-function. 
(5) To expand ' — =f^, r — dx in terms of tetrachoric functions about 
Jo B(p,q) 
the mean. 
The mean is at x = p/(p + q). 
^pq 
The standard deviation is o-= — 
{p + q) + 9 + 1 
Take origin at the mean ; then x =p/(p + q) + ^- Let 
xP-^{l-xr-^ _ 
where D = ^ , y 
dy 
