397 
1907-8.] Inversion Temperatures, etc. 
5. Reinganum’s equation of state is 
_ Rfa; 3 a 
P ~ 
and the expression involving the temperature of inversion is, as given by 
Dickson, 
-p , _ 2 a (v - b) (v - b) (v - b)' 6 (v - b) z 
b vv [2vv - b(v + v)~\ { 2v 2 v' 2 - 2 bvv(v + v) + b 2 (v 2 + v 2 ) } ’ 
From this we obtain 
<« . A . gP '[ ] { } _ A) _ _ (»~ b ) ( 2?) '- 6 ) ,.0 ~ &) 2 \ 
dv 2 a (v — &) 4 (-y - 7>) 3 \ ?; 2W - &(*; + ®') { } / 
Now the quantity [ ] is positive since v and v' are greater than b ; and { } 
is also positive, for 2{ }=[ ] 2 + b\v ‘ ' — v) 2 . Also the third term in the 
bracket ( ) is less than unity, for [ ]>2 v\v — b). Therefore dtjdv is 
essentially positive. 
Again, 
'dp 2a v(v - 6) 4 / _ v_ bv[ ]{ } 
dv b v\_ ] { )• l v - b v\v - by 
+ -OJz - V -A - Klz5K3^zl) + 2bvVfW\\ 
v-b\ b [ ] { } J J, 
the sign of which is determined by the quantity in the large bracket. 
Writing v = kb and taking v' as being large relatively to v and b, the 
quantity becomes 
- W + 1 W- 447c 5 + 66& 4 - 56/c 3 + 28& 2 - 8k + 1 , 
which is negative for all values of k greater than unity. 
Hence the sign of dp/dv is negative under the specified conditions, and 
therefore dt/dp is negative. 
6. Berthelot’s equation 
_ R7 a 
^ v — b tv 2 
leads to the expression 
From this we get 
R£ 2 = 3? ( y ~ ~ b) 
b vv 
2K^ = 3a 
dv 
v' - b 
V 2 V 
) 
which is always positive. Also 
fd'P _ _ 3 a 1 v - b/^ b\ 
dv 2 b vv v -b\ vj ’ 
which is always negative. Thus dt/dp is always negative. 
