398 
Proceedings of the Royal Society of Edinburgh. [Sess. 
7. Callendar’s equation 
P = 
Ri 
v- b + c[ p 
leads to the expression 
bt n = (n+ 1 )ct Q n , 
and so requires a single inversion temperature independent of the pressure. 
8. If we take the equation 
and regard v f as being large relatively to v, the temperature of inversion 
satisfies the conditions 
a n v - b 
R — = I ®(»'- 2)-6(»-l)l: 
dv bn-lv nlK ' v ' 1 
~Rt = — - , 
bn - 1 v n 1 
an 1 
and we also have 
dp 
dv bv n+1 
(v-b), 
which is always negative. 
The case n — 2 is that of Van der Waal’s. The case n = 5/3 is one 
adopted by Dieterici, and it also gives dt/d/p negative. 
9. Thus no one of these six formulse leads to a positive value of dt/dp 
under the specified conditions ; and it can scarcely be supposed that any 
of them are more than roughly inapplicable in the region considered. This 
furnishes considerable evidence in favour of the idea that Olszewski’s 
observations have been affected by a source of error not yet accounted for. 
10. It is of interest to compare the forms which the different equations 
of state give to the expression for dt/dp when v is large. Van der Waal’s 
gives 
dt _ _ b v 
dp R v - b ’ 
which is negative infinite when v — b, has the value — 2&/R when v = 26, 
and approximates to — 6/R when v is large. Berthelot’s gives 
dt _ b v—b 
dp R 2v - b ’ 
which is zero when v = b, has the value — 6/3R when v = 2b, and approxi- 
mates to — 6/2R when v is large. Dieterici’s equation leads to the value 
dt 
dp 
1 v 
2R V^b 
(v + 2b), 
