422 
ME. J. H. JEANS OX THE DISTEIBUTIOX OF MOLECULAK EXEKGY. 
In the same ’way, the variables q, r, and s only enter the integral 1\I, through the 
term D^,Dt, and we may write 
Dt 0 ^ ■ 
Now denotes the rate of increase in f owing to the action of forces of which 
the potential energy is 
Write E = T -f T' + then 
H \ ^ 
\)t drj j d.Tj 
in which p is any one of the p or r co-ordinates, and 77 is the corresponding velocity 
co-ordinate. 
^ Dq 1 dn^ Dri . Dsi 1 dn^ 
jyt ~ \ dqq ’ ’ Di ~ rfr/ 
The right-hand members of these equations will be functions of the p, p\ r, r co¬ 
ordinates, but it is clearly legitimate to put r, r' all equal to zero, and regard the 
expressions as functions of p) and pj' only. 
It therefore ap^Dears that Dy/D^ will be a function of g, r, and s, of Avhich the 
degree is [n — l), and upon integration, that M is a function of g, /’, and s of the 
same degree. 
The terms ^ and X ^ ^ 
ct dt 
and s of degree n. 
§ 25. It is therefore clear that the correct form to assume for y is a rational integral 
algebraic function of the co-ordinates g, r, and i?. 
If we assume y to be the most general function of degree n in these co-ordinates, 
the coefficients being functions of the time, and if we substitute this assumed value 
for y in equation (xvii.), we shall get. on each side of the equation, a function of 
g, r, and s of degree n. 
If therefore Ave equate the coefficients ot every term on the two sides of the 
equation, we shall have found a solution of equation (xvii.). p. 421, inasmuch as this 
equation is noAv satisfied identically for every value of g, r, and s. 
The process of equating these coefficients leads to a series of difierential equations, 
in Avhich the time-rate of increase of every coefficient is given explicitly in terms of 
the other coefficients and of It. If, therefore, Ave suppose the coefficients to A'ary A\’ith 
the time in the manner given by these equations, the A'alue of y so obtained AA-ill be a 
solution of equation (xA'ii.) for all time. Since the equations inA’olA'e h, and h A-aries 
Avith the time, one further equation is required before Ave can express the co-ordinates 
at any time in terms of the initial A'alues of the co-ordinates and the time. This 
additional equation is supplied by the fact that N, the total number of molecules, 
remains constant. 
AA'hich occur in equation (xAui.) Avill be functions of g, r, 
