through Gases between Parallel Plates. 239 



we have n given by the equation 



J 3 (l-2«) 2 =«(«-i); ..... (7) 



and the differential equation becomes 



__ £n(n-l)0 e 5 e . . . . (8) 



Now equation (7) has two roots. If n-y and ?2 2 be these 

 two values, then ?? 1 + n 2 =l; and so unless n x = n 2 — \ there 

 will be two equations of form (8) corresponding to any one 

 of form (4). Regarded merely as a differential equation, (8) 

 may be solved in several cases : — 



(i.) When a = 0. 



.ii.) When e = 2. 



(iii.) When n = ^. 



(iv.) When n = e or 1 — e. 



As regards case (i.), our problem does not allow of a being- 

 zero. 



Further, e is necessarily a positive fraction, and so case 

 (ii.) is useless for our purpose. 



Case (iii.) corresponds to c = 0, or in other words to R] =R 2 , 

 and the equation of conduction under these circumstances has 

 already been solved by Professor J. J. Thomson (' Conduction 

 of Electricity through Gases,' p. 66, or Phil. Mag. vol. xlvii. 

 p. 253, 1899). 



Case (iv.) requires 



*_*-!) or6 _fci), . . (9) 



2c 2 (l-2e) 2 Q--2e) 



But he R, R 2 



2C* ~ {R.-Ry) 1 



Thus we get either 



Ri "1 



or 



e = 



R,+R 2 I 

 R ; + R 2 * J 



(10) 



Now both these values of e are positive fractions, and 

 therefore are possible. For air =^ = 1*22 approximately, 



