252 Mr. J. H. Jeans on the 



These considerations enable us to form some idea of the 

 disposition of the curves in that part of the plane of p, y with 

 which we are concerned. 



Any curve starting from A and meeting OG (fig. 1) must 

 clearly be an arched curve terminating in B For the con- 

 ditions already laid down prevent it from meeting either the line 

 OB or arc GEB of the parabola, or again from bending back 

 and recrossing OG. The limiting curves of this system are on 

 the one hand the axis of p. and on the other hand the curve 

 which actually passes through G. At no point on any 

 curve of this system can y reach a value greater than OG ; 

 y, therefore, is always less than a critical value, y , given by 



'2 



^ = 0G= 2qe*(£+W (U) 



There is a second system, consisting of curves which pass 

 through A but do not meet OG. These must therefore meet 

 AG, be bent away from the axis of y after crossing AG, and 

 finally meet A A'. 



It can be shown that these curves cannot attain to an 

 infinite value of?/ before meeting the line AA'. For we can 

 find the shape of the curves which fill up that part of the 

 p, y plane which lies between the lines AA' and BB', and for 

 which y is infinite. The differential equation of these curves 

 is seen by equation (9) to be 



_i_ ^ A dp = 



Awe k 1 + k 2 dy ^ 



and this represents a system of parabolas which are convex to 

 the axis of p. Since these parabolas cannot be members of 

 that system of curves which Ave are now discussing, we 

 conclude that there must be some point A', in the line AA', 

 beyond which this system never reaches. As the curves of a 

 system cannot cross one another, it must be the curve that 

 actually passes through G which reaches this limiting point A'. 

 There is therefore a maximum limit to the value of y on 

 curves of the second class, and this is given by 



t/i = AA'. 

 There is an exactly similar system of curves passing 

 through B and meeting BG. For this system also there 

 is a maximum limit to the value of y, given by 



y, = BB'. 



The system of curves which pass through B, and do not 

 meet BG, must consist of the curves which meet OG. This 



