580 Prof. 0. W. Richardson and Mr. K. T. Oompton on 



the monochromatic radiation have a constant amount of 

 kinetic energy T . We shall assume that this energy dimin- 

 ishes to T F 2 (r) when the electron has traversed a distance r 

 inside the matter from its point of origin, and in addition 

 that it has to do a definite amount of work P in order to 

 escape from the surface. The energy T of the electrons 

 which originate at a distance r from the point where they 

 escape from the surface is then a function of r only. It is 

 given by 



T = T F 2 (r)-P (1) 



Consider all the electrons which emerge from an element e?S 

 of the surface, and which originate at a distance between 

 r and r-\-dr from the point of emergence. Let be the 

 angle between the radius r and the internal normal to dS. 

 Let \ be the coefficient of extinction o£ the light with 

 distance. Then the number of electrons which originate in 

 an element of volume dr at a depth denned by r and 6 will 

 be e -^ rcos9 Adr, where A is a constant. In addition to 

 losing energy the electrons will diminish in number with 

 the distance traversed in the matter. Let the proportion 

 which disappear in this way in a distance r be 1 — F^r). 

 Then the total number which reach <iS and which come from 

 a distance between r and r + dr is 



•71-/2 



= f ^ Vvcosa A t 2 irr 2 sin 0dr¥Ar) ^ S °f ° d6, . (2) 



J -tt/2 T 



= 27rAF 1 (r)<irdS e~ K ^ocdx 



^[l_(l + V )r^]. ... (3) 



Also dT = T ¥ 2 '(r)dr (4) 



„„ BN 2ttA^S1 Fx(r) ri n , , , xrl , K ' 



llms BT = -VT-^F7w [1 ^ (1 + ^ > 1] ' * ^ 5) 

 The values of T which make N a maximum are given 



b J ^v ~ °' or 



61 e-^=(l + X 1 r)- 1 (6) 



On this view T is determined by r alone, and the correspond- 

 ing values of T are given by solving (6) for r and substituting 

 in (1). The maximum value of the kinetic energy of the 

 emitted electrons is that which corresponds to r = 0, so that 

 when r = Q 



/BN\ _ ttA^S m 



\BT7r=o T o F',(0) KJ 



