254 Mr. J. H. Jeans on the 



(ii.) At all points adjacent to the line ?/=0, the sign to be 



given to l J- in drawing the system of curves remains the 



same as before. The same is true for points at which y is 

 infinite. d 



(iii.) Hence, since -~ must change sign in crossing either 



the curve (a) or the axis of y, it follows that the curve (a) 

 must cross the positive axis of y an odd number of times. 



(iv.) The curve [a] cannot meet the lines AA', BB' in any 

 points except A and B. 



From this we deduce that the new curve (a) consists 

 primarily of an arch through A and B, and secondly, of any 

 number of closed curves either inside or outside the arch. 



If, then, a shaded diagram is drawn similar to fig. 2, the 

 directions of the shading adjacent to the arch through AB, as 

 also the direction of the shading at infinity, must remain 

 unaltered. 



The four types of curves of which the existence has already 

 been discovered will therefore continue to exist in the new 

 graph in the p,y plane. The general appearance of the new 

 graph will, however, depend on whether the curve (a) cuts the 

 axis of y once or more than once . 



For the effect of introducing into this locus new closed 

 curves which do not meet the axis of y, is simply a twisting 

 of the lines already in the graph, and this introduces no 

 essentially different type of curve. But the effect of admitting 

 into this locus new closed curves which do cut the axis of y, 

 or of allowing the original arch to meet this axis more than 

 once, will be the introduction of an entirely new type of 

 curve into the p, y graph, and this will consist of a series of 

 closed curves. Fig. 3 will, I hope, make the truth of these 

 statements clearer*. 



§ 7. The presence of closed curves in the p, y graph imme- 

 diately makes possible a periodic solution of the original 

 equation. For as we pass round such a curve time after time 

 the same values of p and y recur indefinitely ; so that the 

 graph of y in terms of x will be a regular succession of waves 

 and furrows. 



A solution of this type, interpreted physically, would give 

 a striated solution as regards the middle of the tube ; but the 

 whole solution breaks down when we attempt to satisfy the 



* The three closed curves which are symmetrical about the central 

 axis, together with the arch through AB, constitute the curve (a). The 

 closed curves in the p, y graph are seen surrounding the point Z on the 

 axis of y. 



