H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 341 
It remains to determine the formula for ;u,'3,„+2- We have 
- 1 
" 7r{n- 3) r j " ^ ^ c^M^' " 
whence by double integration by parts to reduce the U difierential coefficient we 
obtain 
_ 1 -p^ n-4: + 2-p f « ^ o ^ n - 4 + 2 + 1 ] 
X2p+l,n - ^2 („, _ 3) _ 4) jv'^*' — O + -'P) X2p-l,n-2 " " ^ _ 4 X23J+l,n-2|- 
Putting p = 1 and changing « to n + 2 we have 
n ( n+1 ] . .... 
This may again be read as a formula for [^':j,n+2 '- 
- ^. (i - ^1 + fij^TT^fai (i + « c + 1) (--v). 
Starting with the values of the ju's for n = 3, 4, 25 and 26, the moment co- 
efficients about V = 0 have been determined for n = 5 to 25 in succession. As 
controls the values for n = 20 had already been determined and those for n = 10 
were also obtained at a very considerable expenditure of labour from the very 
slowly converging series of Formulae (xx), (xxi), (xxv) and (xxvi). The initial 
values of the moment coefficients (i.e. those for n = 3, 4, 25 and 26) had to be 
calculated generally to 15 and sometimes to 20 significant figures, owing to the 
numerical factors in (xxviii), (xxxi), (xxxii) and (xxxiv) being frequently greater 
than unity, and thus errors in the last figure being repeatedly multiphed. According 
to the special value of p, it was found best sometimes to deduce moment co- 
efficients of n + 2 from those for n, and sometimes those of n from those for n + 2, 
i.e. to work up from 3 and 4, or down from 25 and 26. It seems unnecessary to 
enter at length here into the many difficulties that arose in the course of these 
calculations. We think they have all been successfully surmounted and that our 
final values may be trusted to the figures actually recorded in the tables. We thus 
found the moment coefiicients and from them the values of and for the ten 
values of p from 0 to -9, and for the values of n, 2 to 25, 50, 100 and 400. 
Diagram I shows that our 270 frequency curves are fairly well distributed over the 
most frequently occurring portion of the ^j, ^2 plane. Now our view is that the 
constants /J^, ^2 describe adequately for statistical purposes the bulk of the usual 
frequencies distributions. But we have provided tables of the values of the 
ordinates for the above 270 curves. Hence hrj interpolation it will now be possible 
to determine rapidly ordinates which will graduate with reasonable accuracy any 
frequency distribution whatever quite apart from the idea of sampling normally 
correlated variates *. 
* Francis Gallon frequently insisted on the importance of foiniing Tables of freciuency ordinates, 
which would graduate any frecjuency distribution in the /Sj, /3o plane. A scheme for covering this plane 
