of Edinburgh, Session 1884-85. 
21 
arranged themselves in the above manner, under the influence of 
the current, and indicating these apparent lines of force. 
The curves presented such an interesting appearance that the 
author thought it desirable to record the observation. 
5. Note on a Theorem of Clerk -Maxwell. By Prof. Tait. 
At the last meeting of the Society Sir W. Thomson again raised 
the question of the validity of Boltzmann’s Theorem, to which I 
had called attention two sessions ago. He expressed, at the same 
time, some doubts as to Clerk-Maxwell’s Theorem (of which Boltz- 
mann’s is an extension) ; doubts, however, confined to the proof 
given by Maxwell, not as to the truth of the theorem itself. This 
theorem is the extremely important one, that in a mixture of two 
kinds of particles the average kinetic energy of the particles of 
each kind is the same. The proof, as given in the Philosophical 
Magazine for 1860, is so very condensed as rather to surprise the 
reader by the extraordinary rapidity with which it seems to show 
that the final average is attained. I have, therefore, expanded it 
so that the nature of the approximation to the average may be 
clearly traced. 
Lemma. — The mean value, of the square of the distance of any 
point on a sphere from an internal or external point A, is the sum 
of the squares of the radius of the sphere and of the distance of A 
from the centre. 
The proof is immediate. Divide the spherical surface into pairs 
of elements by double cones, of very small angle, whose vertices are 
at the centre. Por each pair of these the theorem is obviously true. 
Hence if the speeds of two points be p and q, their mean square 
relative speed is p* 1 + cf. 
Clerh-MaxwelV s Prop. VI. 
The following figure shows points on a sort of hodograph. Let OP 
represent, in direction and magnitude (p), the velocity of a particle 
of mass P. Similarly OQ that of Q, speed q. Let <POQ = a. 
Let G be the centre of inertia of P and Q supposed placed at these 
points in the figure. After the impact G remains undisplaced, 
