( 698 ) 
(ww) = 
4 Ae, cosa’ — Be, cos (a + a). 
4 Aeg, cosa — Bestel. 
2 A £, cos a — Be, cov (a+ py], - 
loy=" 
| 
wo) = — | 
| 
| 
2 A &, cos a — Be, cos (a) + 8) ' 
from which 
s. (oo 
4 Ate’ + B'e,’ —5 ABe, eg) 
(Lw) a A | 4 A e* + Be, — 4ABe, e, cosB ' 
a, 
Therefore 
u, 
3.0 = SAB, &, cos B, 
2 
and similarly 
so = — aA Be, &, cos @' 
3 
The only stable solutions are thus those in which 8 and 9’ are 
both 180°, and the only solution which satisfies this condition is (6). 
This solution, i.e. the case actually occurring in nature, is thus 
found to be the on/y stable periodie solution. 
It needs hardly be mentioned that all the proofs given above 
suppose, that the developments in powers of ¢ and m; converge so 
rapidly, that the sign of the various quantities used is determined 
by their first term. What the upper limits of « and im; are satis- 
fying this condition, cannot be stated without a special investigation, 
but nature teaches us, that for the values occurring in the system of 
Jupiter the solution (6) still exists as a stable solution. 
Physics. — “Contribution to the theory of binary mixtures, XIII.” 
By Prof. J. D. vAN DER WAALS. 
We have considered the closed curve, discussed in the preceding 
Contributions, as the projection of the section of two surfaces, viz. 
Py aw 
==) ene 
dx? dv? 
axis. Let the z-axis be directed to the right, the v-axis to the front 
and the 7-axis vertically. The projection of these sections on the 
other projection planes will now also be a closed curve, in general 
=— 0, constructed on an z-axis, a v-axis and a T. 
en” il 
