250 Mr. J. H. Jeans on the 



In virtue of this assumption q and a will now be single- 

 valued functions of ?/, and they must, as has already been 

 noticed, be positive for all values of y. 



If t^ is replaced by p-f, equation (8.) will reduce to an 



equation which involves p and y only, and which may be 

 written 



1 k^k 2 dp 

 kire k 1 + k 2 dy 



, *k x k 2 / 2 ./l 1\ 16tt¥\ 



= ^ + 32^^ + ^ ^^U " kJ P ~ ~W) * W 



Taking /> and ?/ as coordinates, a graph, may be constructed 

 showing the relation between p and y which is implied by the 

 above equation, and this graph will consist of a singly-infinite 

 system of curves. 



Since y, n 1} n 2 must all be positive, the only parts of this graph 

 which are of any importance in connexion with the present 

 problem lie in that part of the p, y plane which is bounded 



by the two lines p = -=— and p = t-, and is on the positive 



side of the axis ofj?. 



For any given values of p and y equation (9) gives a 



single value for ~, so that through every point in the p, y 



plane one, and only one, curve passes. An exception to this 



occurs at the intersections of the lines p= -r— andp= — t— 



with the line y = 0. At these two points -4- becomes inde- 

 terminate, and an infinite number of curves branch out from 

 these points. Moreover, the axis of p satisfies equation (9) 



at every point, for at all points on it y = and ~- = go . This 



line is therefore a curve of the system, and hence no other 



curve can meet this axis except in the two singular points 



mentioned above. 



dj) 

 The points on the system of curves at which -/- =0 are, 



by equation (9), given by 



