DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
495 
Solution. 
§ 10. The next case we shall consider is that of a saturated salt solution. 
Let w be the mean value of dp/S9 (v constant) at the temperature 6 for the salt, w' 
the value of the corresponding quantity for the solution ; let v be the volume of the 
salt, v' that of the solution, q the mass of the salt, q' that of the solution; the 
mean potential energy of unit mass of the salt, V 3 that of unit mass of the solution. 
Then, if there is no energy due to strain, electrification, &c., the positional part of 
L will be equal to 
6wv + 9u/v' — g'Vj — q V 2 .(35) 
When there is equilibrium this must be stationary, so that, if we suppose a small 
quantity Sq' of the salt to melt, the value of L must remain unaltered. 
If a and p are the densities of the salt and the solution respectively, then when 
the mass Sq of the salt melts the changes in the volume of the salt and the solution 
are given by 
Sv = — Sq'Jcr 
(36) 
As the properties of the fluid may alter with the amount of salt dissolved, we must 
regard w and V 2 as functions of q. Remembering this, we see that the change in 
the value of L, when a mass Sq of the salt dissolves, is 
/I 
dq 
and this, by the Hamiltonian principle, must vanish. Hence, equating the quantity 
inside the brackets to zero, we have 
w , , d 1 
— + wq — - 
P dq p 
/ dw' w 1 / ,dN\ 
■ v iq = r + eP>-^' + q W- 
. . (37) 
This equation would determine p, the density of the saturated solution, if we 
knew how w\ V 3 , and p depended upon the amount of salt in unit volume of the 
solution ; as, however, w r e have not this information, we cannot reduce this formula 
to numbers. We can, however, use it to calculate the effect upon the density of the 
solution of any change in the external circumstances. In the preceding investigation 
we have supposed that the energy possessed by the salt and the solution was all 
intrinsic as it were, and that none of it depended upon strain, electrification, magne¬ 
tisation, and so on. 
