3. THEORY OF STATISTICS. 
random sampling. Since, N being given, fixing the contents of 
any n of the classes determines the = + 1th, there are only = 
independent variables. Let us now suppose that the distribution 
of deviations is normal. Then the equation of the frequency 
“solid ” is of the type set out in equation (15) of p. 331, which 
we will write for the present in the form 
ke ~ 5x2 
x® =a constant, is then the equation of the “ellipsoid” delimiting 
the two portions of the ‘“volume” corresponding to combina- 
tions more or less likely to occur than my, m,, m, . . . . my, 
Accordingly, to find the chance of a system of deviations as 
probable as or less probable than that observed, we have to 
dissect the frequency solid, adding together the elliptic elements 
from the ellipsoid x? to the ellipsoid «0, and to divide this 
summation by the total volume, ¢.e. the summation from the 
ellipsoid 0 to the ellipsoid o. 
In this book we have been concerned with summations the 
elements of which were finite. The reader is probably aware 
that when the element summed is taken indefinitely small the 
summation is called an integration, the symbol [ replacing 2 or iS, 
and the infinitesimal element being written dz. In the present 
case we have to reduce an n-fold integral the summation relating 
to n elements dz, dx,, etc. To reduce this n-fold integral to a 
single integral, the following method is adopted. In the first 
place the ellipsoid, referred to its principal axes, is transformed 
into a spheroid by stretching or squeezing, and the system of 
rectangular co-ordinates transformed into polar co-ordinates. 
The reason for adopting the latter device is that, when twc 
rectangular elements dz, dy are transformed to polar co-ordinates, 
we replace them by an angular element df, a vectorial element dr, 
and a term in 7, the radius vector. When n= such elements are 
transformed, the integral vectorial factor is raised to the n» — 1th 
power and there is an infinitesimal vectorial element, dr, and a 
“solid ” angular element. But as the limits of integration of 
the angular (not of the vectorial) element will be the same in 
the numerator and denominator, these cancel out, while x may 
be treated as the vectorial element or ray. Hence the multiple 
integral reduces to a single integral and the expression becomes 
oc 
( ei po. dx 
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