SUPPLEMENTS—GOODNESS OF FIT. 371 
relation there shown to hold between the normal curve and the 
surface of normal correlation, at once suggests that the same 
principle will apply when there are two variables. 
It was proved on pp. 319-321 that the contours of a normal 
surface are a system of concentric ellipses. Now suppose we 
have a normal system of frequency in two variables z and Ys 
then the chance that on simple sampling we should obtain the 
combination 2’ ' is measured by the corresponding ordinate of 
the surface, and the feet of all ordinates of equal height will lie 
upon an ellipse which will therefore be the locus of all combina- 
tions of z and y equally likely to occur as is z’ y. oF combina- 
tion more likely to occur than 2’ will have a talle ordinate, 
and as the locus of its foot must also be an ellipse, that ellipse 
will be contained within the 2" 3’ ellipse. Conversely, combina- 
tions less likely to occur than 2’ 7’ will be represented by 
ordinates located upon ellipses wholly surrounding the z' #’ 
ellipse. Hence, if we dissect the surface into indefinitely thin 
elliptical slices and determine the total volumes of the sum of 
the slices from az=2' and y=%' down to 2=0 and y=0, this 
volume divided by the total volume of the surface will be the 
probability of obtaining in sampling a result not worse than 
x’ y'; or, if we prefer, we may sum from x=2', y=% to 
z=y=ow, and then the fraction is the chance of obtaining as 
bad a result as 2" 7, or a worse result. 
The reader who has compared the figures on p- 166 and 
p- 246, and followed the algebra of pp. 331-332, will have no 
difficulty in seeing that, when the number of variables is 
3, 4... .m the above principle remains valid although it 
ceases to be possible to give a graphic representation. With 
three variables the contour ellipse becomes an ellipsoidal surface, 
and the four-dimensioned frequency “volume” must be dissected 
into tridimensional ellipsoids; with four variables another 
dimension is involved, and so on; but throughout the equation 
of the contour of equal probability is of the ellipse type (cf. the 
generalisation of the theorems of Chapter IX. in Chapter XII). 
Let us now suppose that if a certain set of data is derived 
from a statistical universe conforming to a particular law, these 
data, & in number, should be distributed into n+ 1 groups con- 
taining respectively ny, m;,, n, . . . . n, each. Instead of this 
we actually find mg, m,, m, . . . .m,, where 
myt+m +... a=nytn +... n,=D. 
The problem to be solved is whether the observed system of 
deviations from the most probable values might have arisen in