410 
then 
The Incomplete Moments of a Normal Solid 
= (f> '{x. z) 
= (/) {oc) .z + x.^{z) 
= — Z + X . xz 
and any combination of operators may be developed in the same way. 
Assume 0" (2) =x. <p"--' (z) - (n - 1) (z) (44). 
Then <^"+' (z) = (f> {x . (z)] ~{n-l)<j) [<f>''-^ (z)} 
= (j) (x) . 0"-^ {z) + x. (z) - (?i - 1) (f)"--' (z) 
= x<f)" (z) - (h) (45), 
which is of the same form as (48). But the relation is true for w = 1, 2, and hence 
is generally true. The solution of this functional equation is 
(z) = - 
n .71 — 1 
n . 71 — I .n — 2 . n — S 
2.4 
z ...(46). 
Writing the factor 
n .n — 1 
x, — 
?i . n — 1 . ?i — 2 . — 3 ,n / N ^ . h, s 
+ x^-^- ... = T,,{x) (47), 
2 . 4 
we may write (jb" {z) = Tn {x) . z, 
and T„ (x) I have called the polychoric function of x of the Jith order. A more 
convenient method of writing it is 
■ft! ft! 2 (ft -2)! 22. 2! (ft -4)! 
In a similar way it may be easily shown that 
2« . s ! (ft - 2s) ! 
..(48). 
ft.ft — 1 ^ „ ft .71 - 1 . ft — 2 . ft — 3 „ , I ^ 
x''-" + ..U^ •••(49), 
2.4 
where ^ = ( ^ + i" ) and f = , e 2 (i -r^) 
\dx dyj 27rVl-?-2 
and it will be found useful to write x" + 
ft . n — 1 
■ + ...= 5H„ {x). The following 
property of these functicms is sufficiently interesting to be noted here. Presuming 
the argument *■ throughout 
T 
T , 
ft ! ' 2 (ft - 2) ! 2- . 2 ! (ft - 4) 
T 
+ ... 
(a) = 
ft! 2(«-2)! 2^2!(ft-4)! 
^ ' ^ 2 V(/i - 2) ! 2 (/! - 4) ! 
(c) + 2^72] (^(^ _ 4) ! ~ 
2^ 2 ! (?i - 6) ! ■■7 
2 (ft - 6) ! 2^ 2 ! (ft - 8) I 
+ 
.(50). 
