n 



990 Sir J. J. Thomson on a 



equation (15). Putting E = E in this equation, we get 

 _ „ _ jM S V2/w 



To get the value of # D — a? B is more difficult ; we see from 

 the form of the equation (16) that the relation^ between 

 E and x could be expressed by means of elliptic integrals. 

 A simpler way is to proceed as follows : if E 2 is the 

 maximum value of E, put E = E 2 — f ; between D and B 

 f is small compared with E 2 . 



Then equation (16) becomes, approximately, 



KD'-M 1 -^)'.- • • '<»> 



since -=-$ vanishes whenE = E 1 (j. e, when f =E 2 — E^. We 



(XX 



see from equation (16) that 



±me (E + 3E 2 ) 

 s/W^fni (E 2 -E ) ' 



If we put £=2(E 2 -E 1 )sin 2 #, then from (17) 



«=2'{?^}V • ■ • > • (13) 



Now at D, f =0 and at B, £=E 2 — E ; we have seen that 

 E 2 — E is much larger than E\ — E , so that at B we may 

 for a first approximation put £=E 2 — E l5 for this value of 



f 2 sin 2 0=1 ; so 6 at B = ~, and by (18) 





We find after some reductions that 



hence a , D _^ = _ j __ j . 



Thus 2(o? D — o? A ), the distance between two maxima for 

 E, is equal to 



L M 



•I "t" i •> 



^2 p 



