350 Proceedings of the Royal Society of Edinburgh. [Sess. 
controllable co-ordinates ; the probability must depend on the state of the 
body, its crystalline structure, chemical composition, etc. 
Dr Bryan, in the paper to which I have referred, says : “ The distribu- 
tion of co-ordinates at any instant is not theoretically really independent 
of the distribution of velocities at that instant. As, however, it is 
physically impossible to control or observe the motions of individual 
molecules, we may assume that the proper measure of the probable distri- 
bution of co-ordinates, estimated according to the best of our knowledge 
of the state of a body, is one which does not depend on the motions of the 
molecules, and remains independent of the time so long as the energy and 
controllable co-ordinates of the system are constant.” This assumption is 
found convenient for mathematical calculation. 
Let us suppose that the co-ordinates of any system are a, b, c, ... and 
the corresponding generalised momenta are p, q, . . . 
A quantity of the form F(a, b, . . . p, q, . . .)dadbdc . . . dpdq ... is the 
probability that the co-ordinates and momenta of the system may be 
between the limits a and a + da, b and b + db, . . ., p and p-\- dp, q and 
q-\- dq, . . ., as defined in Appendix C of a paper read by Dr Bryan at the 
Oxford meeting of the British Association (1894). 
If we multiply a function of the co-ordinates and momenta by this 
quantity and integrate, we obtain its mean value. 
(3) The system (or molecule) is defined as consisting of two particles at 
A and B of masses m x and m 2 . Let G, the centre of mass, be (as, y, z), and 
let AB = y, and for the six co-ordinates of the system let us take x, y, z, 
r, 0, so that the co-ordinates x v y v z x of A and (os 2 , y 2 , z 2 ) of B are 
given by 
x 1 = x + 1 \ sin 0 cos <f> 
y 1 = y + r 1 sin 6 sin <f> 
z 1 = z + i\ cos 6 
x 2 = x + r 2 sin 0 cos <f> 
y 2 = y + r 2 sin 6 sin cf> 
z 2 = z + r 2 cos 6 , 
where 
m i r i = - m 2 r 2 > 
L = 
m 1 + m 2 ’ 
- m 1 r 
m 2 + m 2 ‘ 
0 and (j) are the angular co-ordinates of the straight line AB. 
x 1 = x + r l sin 0 cos </> + r l cos 0 cos cf)0 - r 1 sin 6 sin </></> , 
ij\ — V + L s i n 0 sin + cos ^ s i R ^ + 7 \ s i n 0 cos , 
z 1 =z + ?\ cos 0 -r l sin 00. 
x 2 , y 2 , and z 2 are obtained similarly. 
