of Edinhnrgli, Session 1882-83. 
229 
measured on x, pressures in atmospheres on g, and density on 2 , 
Then the section hy the plane g = 1 is a curve with a maximum at 
x= 4. But from this section the surface rises, in the direction of y 
dz 
positive, faster for lower values of x — for y = l becomes 
greater as x diminishes). Hence it is clear that a proximate section, 
say for ^ = 2, has its maximum ordinate for a value of x less than 
4° C. 
In fact, if e be the expansibility of water, it can be expressed as 
a function of the temperature and pressure alone, Le., 
« =/(<. p) ■ 
Hence simultaneous changes of temperature and pressure, which 
leave the expansibility unaltered, are connected by the relation 
The maximum density point is a particular case of this, for there 
the expansibility is zero. 
How, if V be the volume, we have 
so that 
But 
ii 
dp 
(logv) is the compressibility, and diminishes with in- 
crease of 1. Thus ^ is essentially positive. But so is ^ so long 
as the temperature is above the maximum density point. Thus 
Sp and Si in the above equation have opposite signs. 
I have learned quite recently, and by mere accident, that the 
lowering of the maximum density point of water has been already 
pointed out as a theoretical result by processes essentially the same 
as that just given; first byPuschl* in 1875, then by Van der 
Waals t in 1876. Both authors refer to experiments made by 
Grassij; for Eegnault in 1848, on the compressibility of water 
* Kais. Ac. d. Wiss. Sitzb., Ixxii. 283. 
t Archives Neerl., xii. 457. 
t Annales de Chimic et de Physique, ser. iii. t. 31, 1851. 
