444 Dr. W. F. Gr. Swarm on the Expression for 



The flow per square centimetre through MP due to the group 

 in question is 



_ nv s'm 6 cos ^dO _ E/A 

 (An)l -47rR 2 Xsin^^ - 6 ^ ' ' • ( 3 > 



Now the equations of motion of the electron are 



x = vtcos0+~t 2 , (4) 



y = vt sin 0, (5) 



where £ is the time taken to go from to E. 



Now 



c ^f=J^fTi ( 6 ) 



Substituting from (4) and (5) in (6) and neglecting second 



order terms (i. e. terms of the order ( — \r ) ), and remem- 



Xe \ mv / 

 bering that in terms multiplied by — -g, vt maybe replaced by 



#/cos 0, and that in such terms may be substituted for <fr r 

 we readily obtain (see Appendix, problem 1) the result 



^ = { 1+ S tan2 *} C0S ^ • • • ^ 



On differentiating, we readily find 



cos 



sin 



y^=|l-^(l + itan 2 6')|sin(9^. . (8) 



Hence, since may be replaced by i/r in terms multiplying 

 ~Xe/mv 2 (3) becomes 



(An)l= i^{ 1+ S( 1+ * tan2 ^} «»*••-***•■ (9) 



It will now be convenient to take E as the origin, ty becomes 

 the angle made by the radius vector to the point with the 

 normal to MP, and the total number of electrons 8n reaching 

 E per square centimetre per second from the ring of volume 

 27rR 2 sin yjr dyfr dR passing through and parallel to the 

 plane PM is 



5n=^|l+ ^(l + itan 2 ^)lsinA/rcos^6- R / A ^d!R. (10) 



* In the expression e~" R /\ B, should, strictly speaking, be replaced by 

 the length of the dotted line OE, hnt it is easy to see that the correction 

 on this account is a second order quantity. 



