399 
1907-8.] Inversion Temperatures, etc. 
which is negative infinite when v = b, has the value — 46/R when v = 2b, 
and approximates to negative infinity again as v becomes large. Its 
smallest numerical value is b(2 + J 3) /U. When v = 10b, the value is about 
double the minimum. When t> = 1006, the value is fully 14 times the 
minimum. 
The equation of Clausius gives 
dt _ _v -b (y - b) a + (a + b)(3v + 2a) 
dp 4R * (v-b)( 3v + a) + (v + a)(3v + 2a) 
This vanishes when v = b, is always less than (v — fr)/4R numerically, and 
approximates to — (4a + 36)/24R as v increases without limit. 
Reinganum’s equation leads, in the previous notation, to the expression 
dt _ _1,,,_., 3 6& 3 - 6& 2 + 3& — 1 
dp R 1 ' Add - 16& 6 + 44& 5 - 66& 4 + 56& 3 - 28& 2 + 8& - 1 ’ 
This vanishes when k — 1, that is when v — b. It has the value — 58/787R 
when k = 2, -2784/3713R when k = 3, - 32292/31327R when k = 4, and 
approximates to — 3/2R as k tends to infinity. 
11. Experiments made upon inversion temperatures, when a gas 
expands from a measured volume to a volume large in comparison with it, 
may lead to a decisive test amongst various equations of state. 
In Van der Waal’s equation, the constants b and R have, respectively, 
the values 1’528 and 2’835 when the unit of pressure is one atmosphere 
and the unit of volume is 1 c.c. Hence the fall of the inversion temperature 
per atmosphere of increased pressure is about half a degree centigrade 
when v is large, and increases to about 1° C. when v = 2b. 
Taking 0, the critical temperature of air, as 133° C., and calculating 
p and v from the above value of b along with a = 1682, we get 
p = 1682/2 7(1 '528 ) 2 , ^ = 4*584. Using these values of the critical constants 
to calculate R in Dieterici’s equation of state from the condition R = 8pD/3 0, 
we find that this equation gives dt/dp— — 2 0, 35 when v = 2b, and that it is 
equal to — 3°*12 when v = 106, increasing without limit as v increases. 
Similarly, calculation of the constants in Berthelot’s equation from the 
conditions b = v/ 4, a— 27R 2 d 3 /64p, R = 32vp/9fi, gives —dt/dp roughly equal 
to \ of a degree centigrade, increasing to about J of a degree centigrade 
when v is large. 
Again, as above, Callendar’s equation requires that dt/dp should be 
zero. 
(. Issued separately July 22, 1908.) 
