R A. Fisher 
509 
For the variables x and y it is now necessary to substitute the statistical 
derivatives determined by the equations 
n 
nrfiifi2 =^{oG-x){y- y), 
1 
and it is evident that the only difficulty lies in the expression of an element of 
volume in 2n dimensional space in terms of these derivatives. 
The five quantities above defined have, in fact, an exceedingly beautiful 
interpretation in generalised space, which we may now examine. 
3. Considering first the space of n dimensions in which the variations of x 
are represented, the mean and mean square error of n observations are determined 
by the relations of P, the point representing the n observations, to the line 
tt/i — t^2 — "^3 
... — X-ii , 
for the perpendicular PM drawn from P upon this line will lie in the region 
Xi -\- Xii -\- . . . -\- — nx, 
and will meet it at the point M, where 
/y> — /yt /y> ry^ ly — rf* • 
iX/J — <Aj^ lA/Q lVj • t • t-t/ y 
further, since, PM^ = {x, - xf + {x.^ - x)^ + . . . + {xn - xf, 
the length of PM is ^]n. 
X3 
An element of volume in this n dimensional space may now without difficulty 
be specified in terms of x and /^i ; for, given x and /Xj, P must lie on a sphere in 
n — \ dimensions, lying at right angles to the line OM, and the element of 
volume is 
G^-^^~"'dix^dx, 
where G is some constant, which need not be determined. 
65—2 
