* 
ON THE PRODUCT-MOMENTS OF VARIOUS ORDERS OF 
THE NORMAL CORRELATION SURFACE OF TWO 
VARIATES. 
By K. PEAKSON and A. W. YOUNG. 
(1) In several recent investigations we have found it desirable to have the 
values of product-moment coefficients about the mean of the normal correlation 
surface. The present paper deals with the case of two variates. If the correlation 
surface be t ■,/■>.-, 
Z= e 2 1-')-"V< (r^.o-„ ' <ry /-x 
277-a, a/1 — ^ 
^OyVl — r 
where ct^. and Oy are the standard deviations of the two variates x and y and r their 
correlation, then we define the sth-rtii product-moment coefficient to be 
2 r+co /■+!» 
= ^ xM/zdxdy (ii). 
• - — CO - CO 
Further we write t = 5',,, i/(a',a',) (iii), 
so that Ps, t is a purely numerical quantity and a function of the variable r only. 
Clearly from the symmetry of the surface 
2^2s, 2*+l = Vts+l, 2t ~ 0- 
We are accordingly only concerned with cases in which s ^ t is, even. 
We propose therefore first to give the general algebraical expressions for the 
lower values of fs, 1 1 ^"^^ secondly to provide tables for the numerical values of 
these product moments proceeding by increments of -05 in r. 
Since s + i must be even if < be not zero, it follows that s and t must either 
both be even or both be odd. In the former case pig, t does not alter when r changes 
sign ; in the latter case ji^^ f for negative r is simply ^J.,, t for positive r with the sign 
changed. It is accordingly only needful to table ^ for positive values of r. 
For the purpose of testing computations the following formulae are of value: 
Ps,t= (s + t-l) m-i,t-i + (s - 1) - 1) (1 - r2) iV2,<-2 (iv), 
Ps,t = ('-!) lh,t-2 + srPs-i,t-i = (s - 1) Ps-2,t + fr2?s-i,t-i (v). 
Or, again we may write 2\ * = (^'''Y'^s.t (vi), 
and we have 77.,, < = (/ ^ 1) 77^^ + tt^.^^ (vii), 
which is capable of numerical evaluation in a single machine operation. 
The general values for any normal product-moment coefficient are 
r(2s+ 1 )! (2^+1)!^-^ ( (2?f« 1 
?2.+i,2m- 2^+*' ~-\%l^s-u)lit-u)l (2«+l)!} 
