-MR. J. II. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
423 
We have (see § 23, equation (xiii.) ), 
N = ^ HF c/P = j e-M^c^+ 2 ^) j-Y dp dF du . . 
. (xviii.), 
and the equation dN/cZ^ = 0 is the equation required. With the help of the other 
equations, it can be written so as to give dhjdt explicitly as a function of h and the 
other co-ordinates. 
Thus if we have the initial form of y given, we have obtained sufficient equations 
to enable us theoretically to determine y at any subsequent time. It is not proposed 
to attempt the solution of the system of equations in the most general case ; the dis¬ 
cussion is confined to the modified forms which these equations assume in the two 
states of which the physical interest is greatest, namely the steady state, and the 
state in which the gas is non-luminous. 
O 
Solution in Steady State. 
§ 20. The mathematical condition that a steady state may be possible, is that it 
may be possible for the time rates of variations of the coefficients to vanish 
simidtaneously. From the equations found by equating to zero the time rates of all 
the coefficients except h, it is possible to find these coefficients in terms of h so that 
the condition for a steady state to be possible is that the function of li obtained by 
substituting these values in the expression for dh'dt shall vanish identically for all 
values of A. It is, however, known that the condition that a steady state shall be 
possible is that G shall be identically zero, and we may therefore begin by putting 
G = 0 and neglecting the equation dhjdt — 0. 
Thus all the equations necessary are contained in the characteristic equation 
satisfied by y, and this is now {cf. equation xvii., p. 421) 
H 
or substituting for d^jdt from the scheme of p. 419, 
) (L + M) 4- i (? s, - h 
II ' 1 ' crj fq oq 
= 0 
(xix.). 
Let us assume as a possible value for y the most general expression of degree n in q, 
r and s, the coefficients now being independent of the time. 
Consider the system of equations which is obtained when we equate to zero tlie 
coefficients of terms of degree n in equation (xix.). The terms of degree n in the 
integral L arise entirely from the terms of degree n in y (see § 24). These terms 
