if this point is one where for the curve has a maximum or

The area between the curve and the lines =X], X=X2,

8=0, is

Hence where

dx

3r-2

dx ୪

=0, i. e. tangent is parallel to axis of æ,

3 ; and eliminating y between this equation and (a) we have

-75

.5

·25

-251

.. It has a tangent parallel to the axis of a when = or 1, and has two inflexions where

a=1+, i. e. 1.577 and 423, approx.;

3

2.5

Fig. 3.

7.5

10

it is of degree and class 4, has two cusps and a double point all at infinity along x=0, has one double tangent (the axis of a, which touches it at infinity, and where =), and is of zero deficiency. Its radius of curvature at any point is

the area enclosed between it and the lines =X2, X=X1, and

6=0 is

It is shown in fig. 2.

As in the case of the isothermals, so in the two families treated of here, we have confined ourselves to a mathematical discussion, and have left out physical considerations; as before, the only part of the curves which has any physical meaning is that for which >, 0>0 (and .. y>-27), and the only part for which the physical interpretation of the equation necessarily holds is that for which >, 0>0 (and y>-3).

Instead of considering van der Waals' equation as represented by three families of curves, we may consider it represented by the surface of the fourth degree,

3yx3 −(y+80)x2+9x−3=0.

(a)

This is a ruled surface traced out by lines parallel to the plane of 0, y.

The line of striction is the intersection of (a) with the

cylinder

(y+80-9)+81(y-40+3)2=0.

The plane =0 meets the surface in no finite point; the plaue = meets the surface in a line parallel to the axis of y. Two sheets of the surface touch one another along the line at infinity in a=0, which is parallel to y+80=0; the surface also contains the line at infinity in the plane y=0 which is parallel to a=0; and the line at infinity in the plane = which is parallel to y=0. The normal at the point a', y', e' is

3

The tangent plane is therefore parallel to the axis of a when 46'x'=(3x-1)2

and

·· yd3=3a'—2 and (y/' + 80′ −9)3 +81 (y' — 40' +3)2=0. At the point (1, 1, 1) the principal radii of curvature are