or 
136 Mr. J. D. Hamilton Dickson on the 
It will be necessary to get the equations to these isopiestics 
Eliminating <f> from (24), after a few steps we have 
729a? 3 - 216^ + (729 - 27a> 2 - Uy 2 
+ 108a^-32a#-f(108a-4a 2 ) = 0. . . (25) 
We shall also require the envelope of these isopiestics ; 
its equation, following the usual process, is immediately 
16(729tf 3 -216^ 2 z/ + 729^ 2 -64?/ 2 ) 
+ (27^ 2 -108# + 32 < y-108) 2 = 
(* + 2) 2 {(^ + 2) 2 -|^}=0 . . . (26) 
showing that the envelope is a parabola. This envelope is 
the locus of points representing an expansion between 
pressures differing only infhiitesimally in a Joule-Kelvin 
experiment, and (for the present) appears to be independent 
of the magnitude of the pressures, but we shall see later that 
only a finite portion of the curve is involved in the question, 
and also a limited range of pressures. We ought to arrive 
at the same result from Porter's equations (pp. 555, 556 of 
his paper). Writing them in the present notation they are 
thus verifying equation (26). 
These isopiestics and their relation to the envelope are 
shown in fig. 1, PI. VIII. The zero-isopiestic consists of the 
straight line Oo>, and the parabolic arc Q\{r, whose equations 
are 
21x-fy = (27) 
and 
27* 2 + 27.z + 8?/ = (28) 
The envelope VMLN is touched by this isopiestic twice, namely, 
at the points M, N. As a increases from zero the successive 
isopiestics each touch the envelope at two points, the one 
leaving M, the other N, and approaching each other towards 
L. These points of contact are given, for the a-isopie&tic, 
by the equations 
*+2=±|(«±6), y=^±^,. . . (29) 
where u 2 — 9 — a. Hence we see again that a cannot exceed 9, 
