446 Dr. W. F. G. Swarm on the Expression for 



first sight true on the line of argument that the electrons 

 are on the average in the middle of their journey \. The 

 assumption is, however, as we shall see, not true, and is one 

 of the causes of discrepancy between (1) and (2). 



Now returning to (10) and omitting the factor €~ R/ \ we 

 have, for the number (8n)i of electrons which have suffered 

 collisions in the element specified by <ffi and d-^r, and which 

 would pass through 1 square centimetre per second at E if 

 they had suffered no more collisions, 



(S«)i=^{l+ |£(l+ita*+)}rfn + ooB+i**da. (12) 



The number (Sn) 2 of the above set to be found in 1 c.c. 

 at E is obtained by dividing this by .{% the velocity (parallel 

 to the x axis) which the electrons of the above set attain by 

 the time they get to E. In the Appendix, problem (2), it is 

 shown that 



• jt\.ex ^±ex 1 



x = v -j 1 H « + -. — 5 tan 2 sir > cos ijr, 



L inv* 2mir r J T 



so 



that 



71 



[Bn) 2 = ^r-sin yjr . dR d-dr. 



Now the number of the above set which travel a distance 

 between r and r + dr is 



^- 2 €~ '^dr.dR sin ^dyjr (13) 



From the whole range to it/2 the number is 



n 



-r^dr . dU I sin ^ d^= ^ e-*dr . dU ; 



and taking account of the electrons which have come from 

 the right-hand side of the plane, we have to multiply by 2, 

 so that we have for the total number (Bn) z of electrons to be 

 found per c.c. at E which have come from the shell of 

 thickness dR, and which are in the process of travelling 

 distances between r and r + dr, 



(8n) 3 = ^- 2 e-^drdK (14) 



A, 



Now keeping our attention on the electrons which travel 



