488 



Dr. "VV. F. G. Swarm on the Electrical 



Fiar. 6. 



Appendix. 



Problem (1) in reference to the equations involving the 

 quantity ij on page 477. 



Let the ordinates of the curve AOE (fig. 6) represent the 

 pull towards the left, on an electron, due 

 to the molecular forces. AM and NF 

 represent the faces of the gap. Different 

 positions across the gap are represented by 

 different points on BG, being the centre 

 of the gap. Let the dotted curve DPF, 

 which is obtained by subtracting the con- 

 stant pull due to the electric field from the 

 curve AOE, represent the resultant pull 

 on the electron when the field is acting. 

 The place of zero pull is no longer but 

 P when the field is acting. 



In order ihat an electron shall be able to cross from B to 

 G it is necessary that it shall have enough energy to get to 

 the point of zero force. When there is no field this energy 

 is ^mu 2 , and is represented by the area AOB. When the 

 field acts the energy necessary is represented by the area 

 DPB. We shall denote an area by putting a line over the 

 top of the letters. Thus the limiting value of \mx 2 is 



mi' 2 = DPB = AOB-AOTD + POT = imw 2 -LROB + POT 



Ye 



=i mw 2_ _L?+P0T, 



for it is to be noted that AOTD = LROB, since by drawing- 

 vertical lines through these two areas we divide them up 

 into a number of corresponding elementary parallelograms, 

 each pair of which, being on the equal bases and between 

 the same parallels, are equal. Writing 7]e for the area POT 

 we have 



• 2 2 V* , 2 V e 



An exactly similar argument applied to the electrons flowing 

 from right to left leads to the result 



2 — „,2 



Ye 2rje 

 v?+ -+ — ■ 

 in m 



It will be noted from symmetry that the curve AOE must 

 have either a single point of inflexion at 0, or two points of 

 inflexion, one on each side of 0, and it is thus probable that 



