Resistance of Thin Metallic Films. 491 



the usual way small quantities of the order — ^,we readily 

 o btam 



COSl/r= 1 + - !_ )(1 + _J_ COS COS0. 



r \ 2mc cos 0/\ mc I 



Putting ct cos # = t v in terms multiplied by 'X.ie/m we find 



cosijr=(l+?^tan 2 0)cos0. . . . (16) 



Differentiatino-, we readily obtain 



sin i|r. <fy= 1 1-^^(144 tan 2 0) j sin 6cW. (17) 



Hence, since 6 may be replaced by -v^ in terms multiplied 

 by —\ (12) becomes 



An = i^k* i 1 + S? ^ + * tan= *> } cos + e ~ m 8 "- (18) 



It will now be convenient to take E as origin. yfr becomes 

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

 normal to PM, and writing $n=Ae~ hm * i c 2 dc, the total 

 number of electrons reaching E per square centimetre per 

 second from the ring of volume 27rR 2 sin -v/r . dsfr . dR passing- 

 through and parallel to the plane PM is 



(An)x= ^e-'^VVc/l-t- Xl T(l + itan 2 >/r)T> sin >/rcos f e - R > df.dR. 

 1 J .... (19) 



To obtain the current density ? L corresponding to the flow 

 from left to right we must multiply this by e and integrate 

 for all the electrons which succeed in getting across the 

 plane. 



If .i; is the x component velocity with which an electron 

 reaches the gap, then in order that the electron shall be able 



'Ive V<? 

 to cross, it is necessary that or > vr-\ _ - ' as was 



pointed out on p. 477. Now from (13) m m 



x = c co> 



e + 2±t-e(l + - ■>"'■':, )co S 0. 

 m \ mc* cos- v) 



Replacing cos 6 by its value in terms of cos ifr, as found 

 above, we have, on substituting yjr for 6 in the terms multi- 

 plied by X^/mc 2 , and neglecting second-order quantities 



^-fcosf (1+— -, +- r i -^tair\H. 

 V mc 2 zm<? • J 



