138 Mr. J. D. Hamilton Dickson on the 
for such a temperature two isopiestics must intersect. I 
distinguish between an inversion-point and an inversion- 
temperature ; and it will be convenient to refer to the two 
isopiestics passing through any inversion-point as the higher 
and the lower according as the corresponding a has a greater 
or less value. An examination of fig. 1 will lead to the 
following conclusions, which have been verified by calcu- 
lation. All points of intersection of any two isopiestics lie 
within the curvilinear triangle MONLM ; hence inversion 
points only exist within, or on the contour, of this triangle. 
The highest isopiestic which can take part in determining 
an inversion-temperature is a = 27, for this isopiestic cuts 
the triangle only at the point 0. Any isopiestic between 
this and a = 9 may be the higher of the two isopiestics of 
any inversion-point. Any isopiestic from a — 9 to a = 0, 
inclusive, may be the lower of the two required ; and no 
isopiestic above a=9 can be the lower one. Every lower 
isopiestic (t. e. one whose a<9) can only have associated with 
it as the higher isopiestic, one from a limited range of iso- 
piestics extending from itself to a particular value of a, 
depending upon the lower isopiestic, and less than 27. Thus, 
in fig. 1, it is easy to see that the range for u=l extends to 
about 22 — the last being that isopiestic wdiich touches a = l 
and lies not quite half-way from 20 to 25 ; similarly for 
a — 5, the range gets to about 14. No temperature can be 
an inversion-temperature which exceeds 6f times the critical 
temperature. For every temperature below this value^ any 
point on the horizontal line representing that temperature, 
and lying within the curvilinear triangle, gives a pair ot 
pressures which have this temperature for inversion-tempera- 
ture. Thus in order that three times the critical temperature 
may be an inversion-temperature, we may begin with a 
pressure of about 21^ times the critical pressure and expand 
into a vacuum ; or we may use an initial pressure of 20 times 
the critical pressure and expand down to the critical pressure ; 
or finally we may employ two nearly equal pressures about 
9 times the critical pressure. 
The equation (25) for the isopiestics may be regarded from 
other points of view, each of which will, on occasion, be 
preferable, as helping us to see more clearly the limitations 
of the question. It expresses a relation between x,y, and a — 
potential, temperature, and pressure. By making a constant 
we have already got the equations of the isopiestics in the 
temperature-potential plane. If we make x constant, we 
shall have the equations of the isopotentials in the temperature- 
pressure plane ; and if we make y constant, we shall have the 
