256 Mr. J. H. Jeans on the 



We have seen that if condition A is fulfilled all the curves 

 which constitute the p, y graph come under one or other of 

 the four classes represented in fig. 2. 



We shall in the following sections confine ourselves to the 

 consideration of these four types of curves, and it will appear 

 that if these four types alone exist, there is only one type of 

 solution possible in addition to that already arrived at by 

 Thomson. 



Even if condition A is not satisfied this alternative type 

 of solution will still be possible, but, as has already been 

 pointed out, we are no longer justified in asserting that it is 

 the only alternative. A second alternative becomes possible 

 on assuming that the gas near the electrodes is in an 

 abnormal state. 



§ 8. From our knowledge of the graph in the p, y plane it 

 is easy to deduce the corresponding graph in the plane of a?, y. 

 An intersection of a p, y curve with the axis of p will give 

 rise to a maximum or minimum value for y on the corre- 

 sponding x, y curve, and an intersection with the curve (a) 

 will give rise to a point of inflexion. The new graph, in 

 strictness, consists of a doubly infinite system of curves, but 

 it will be sufficient to consider the system as singly infinite, 

 a single curve corresponding to each curve in the p, y plane. 

 A second infinity is then obtained by moving each curve 

 parallel to itself along the axis of oo. 



The curves in the plane of #, y fall into four classes 

 corresponding to the four classes of curves in the p, y plane. 



The curves of the first class will each give rise to a single 

 arch in the plane of x, y ; this arch stands on the axis of #, 

 and is on the positive side of this axis. Since we are now 

 considering the case in which q and ex. are no longer constants, 

 this arch may or may not have points of inflexion. Every 

 curve of this system leaves the axis of x at an angle 



tan -1 (-r- J, and meets it again at an angle tan - \~j~ )• 



The limiting curves of this system are : — 



(i.) An arch which shrinks to a point on the axis of <r. 

 This corresponds to the limiting curve AFB in fig. 2. 



(ii.) An arch of infinite span which touches asymptotically 

 the line y=yo (see equation 11). This corresponds to the 

 curve AGB in fig. 2, and y must now be defined as the 

 smallest (or, if condition A is satisfied, the only) root of the 

 equation 



y= iqe\K + k 2 r • ' • * ' (12) 



