TO THE THEORY OF EVOLUTION. 
19 
zSxSij = ^ Sx'Sy' expt. j - + */' 2 )}- 
Now give the area a uniform slide parallel to the axis of x defined byr/^/1—r 3 
at unit distance from that axis. This will not change the basal unit of area 
Sa = §x'Sy', and analytically we may write 
X = x' - y’r/Vl ~r\ Y = y', R 2 = X 2 + Y 3 . 
Whence we find 
zSxSy = 
_N 
2ir 
8a expt. (— |Tt 2 ). 
This is the mechanical changing of the Yule-Brill model analytically represented. 
The surface is now one of revolution, and the proof would have been precisely the 
same if we had written in the above results any function/, instead of the expo¬ 
nential.^ It is easy to see that any volume cut off by two planes through the axis of 
the surface is to the whole volume as the angle between the two planes is to four right 
angles. Further the corresponding volumes of this surface and the original surface 
are to each other as unity to the product of the two stretches. Lastly, any plane 
through the 2-axis of the original solid remains a plane through the 2-axis after the 
two stretches and the slide. These points have all been dealt with by Mr. Sheppard 
(p. 101 et seq., loc. cit.). I will here adopt his notation r = cosD, and term with him 
1) the divergence. Thus cot D is (in the language of the theory of strain) the slide, 
and D is the angle between the strained positions of the original x and y directions. 
Now consider any plane which makes an angle y with the plane of xz before strain. 
Then, since the contour lines of the correlation surface are ellipses, the volumes of 
the surface upon the like shaded opposite angles of the plan diagram below will be 
equal; and if they be n 1 and n 2 , then n x + n. 2 = -^N. If and n 2 ' be the volumes 
after strain, then by what precedes we shall have 
n x — cr 1 cr 2 v / 1 — r 2 X n/, n 2 = oqo-oV 7 1 — r 2 X n 2 ', 
and (n 2 - + nd) = « - <)/« -f n 2 '). 
* The generalisation is not so great as might at first appear, for I have convinced myself that this 
property of conversion into a surface of revolution by stretches and slides does not hold for actual cases 
of markedly skew correlation. 
D 2 
