402 
The Incomplete Mo?nents of a Normal Solid 
In the course of the analysis it was found necessary to employ functional operators 
which lead to the development of functions of the form of 
n .n—\ „ ?i . ?i — 1 . ?? — 2 . ?i — 8 
X ^ X ^ - ^ •••• 
These in a somewhat ditferent form have been termed tetrachoric functions and their 
values tabled {Tables for Statisticians and Biometricians, Cambridge, 1914). In 
order to avoid confusion with these tabled functions and also to suggest their con- 
nection with multiple celled tables I have tentatively called the functions developed 
in the present paper, polychoric functions. The incomplete multiple product moment 
is found to be expressible in a form of mi;ltiple polychoric function which is itself 
reducible to a function of single polychoric functions. 
2. Notation. 
As the integrals in the following paper are very complicated unless an abbreviated 
notation is used, it will be necessary to resume some of the well-known formulae 
connected Avith the normal equation in order to be intelligible and to avoid con- 
fusion. 
Consider the surface represented by the normal equation 
XT _ -r'^g-,.^ + n"^l<T,r - '2r.v'i/'/(r,(r,f 
Nz'{x',y') = - 7t=^«' ^^'-''I (!)• 
lirar^.a„ v 1 — 
If we write — — x and — = y and . ■ 
]Srz(x,y) = -—==y Mi-r^) (2), 
27r V 1 — r- 
then = (3). 
If now the surface be cut by the plane x' = h' then the area of the face of the 
section will be 
N\ z'{h,y)dy' = ^^ =— - (4). 
If another plane be drawn at y' = k' it will in a similar manner expose a surface 
whose area is 
N z' (x', k') dx' = = ^ (5). 
./ -00 v27ro"jy 
The two planes will divide the volume enclosed by the normal surface into four 
quadrants a, b, c, and d as in the usual tetrachoric scheme 
a 
b 
c 
d 
in which the x's go from — oo to 4- oo in the direction of ({ to b and y's similarly 
from a to c. The a quadrant will be considered as the leading or standard quadrant 
