406 
The InGomplete Moments of a Normal Solid 
The integrals are known and the ms can be found by solving the equations. With 
high moments the equations become cumbrous and it is difficult to follow the 
relations between the moments. A more powerful method has therefore been 
adopted. 
(2 \ant 
— j of an n-fold normal correlation surface by means 
of functional operators. 
The rationale of the process to be developed in this paper will be best under- 
stood by considering at once the general case. The details for a normal surface in 
two variables will be worked out in a later section. 
Consider the equation to a normal correlation surface 
_ SAii.ri- + 2SAi2.riJ2 
Nz{x^, x^, ... Xn) = ^ e 2A 
(27r)2 VA 
•(24), 
where as usual 
.(25), 
and A„, Ajo are the co-factors of ?-„, r^, ... and ?•„ = r.^ = ... = \, and ?V3 = ■^21 etc. 
The general problem is to find the Zi, l^, 1^ ... ^^th product moment of the 
of the form, that is to reduce the integral 
. nij^ ,^ = iV ... x/' xj"- . . . Xn''' z (x^ , x^^... x^) dx, dx^... dxn 
J ~ CO J ~ cc J ~ <x> 
... (26) 
Differentiating partially by 
.(27). 
dz _ 
1 
dxj 
A 
dz _ 
1 
dx.. 
A 
dz _ 
1 
dxs~ 
~A 
Hence remembering that A^t is the co-factor of r^t we have 
' dx.. 
dz_ 
'' dXn. 
= - -^{Ax,)z = -XiZ 
