60 Generalised Pi'obable Error in Multiple Normal Correlation 
The ellipsoid in ?i-fold space can be reduced to a sphere by proper squeezes in 
the directions of its principal axes, and accordingly its volume can be found from 
that of a sphere in ?z-fold space. The volume of a sphere in ?i-fold space 
2"a"(V7r)" 
1.2.3 ... \n 
, if 11 be even, 
= — - — ^^~r- — - — , if n be odd. 
1 . 3 . ...n 
Hence the volume of an ellipsoid in ??-fold space 
= 1 \ Q 1 ' if be even, 
1.1.6... -kn. 
■ ^""'■'■'"'"■■■'-'^">'-',if„beodd. 
1.3.5 ... n 
Applying these to our special cases we have for the volume of correlation 
ellipsoid, V: 
V= \ "A, \v — if n be even, 
1 . 2 . 3 . . . 
= ^^'""^-^'-^-%"<'^->-, if „ be odd. 
1.3.5...« 
Thus for the two cases : 
^Y=—^^^ — 'VA:^'—^ (?i even), 
and = 2^'"-''2.v.^..W-^ ^^d). 
Now the total amount of frequency between two contour ellipsoids is 2^8 F, and 
accordingly the frequency inside a given contour ellipsoid x is given by : 
J [\dV= ^i-T^ 9 f ^ " " 
^ j „ 2 . 4 . b . . . /; — i j 0 
1 P _ I " 7 
Let («) = -TT= ^'"^ '■''^ 
be defined as the " incomplete Hth normal moment function." Then 
J V27r^._,( xL be even, 
^' 2.4.6 ...?i-2' 
if be odd. 
1.3.5 ...?i-2' 
Now /^/iV is the chance of an observation falling inside the contour x> and 
l-IJN the chance of its falling in the fringe outside this contour. 
