640 BELL SYSTEM TECHNICAL JOURNAL 



Those which escape are those for which \mu- exceeds eb\ we obtain 

 the mmiher of them by integrating (6) from u = \/'2eb/m to u = <x>, 

 and find 



I, = 7iVkr 2wm e-**^. (8) 



This, siii)|)osing )i and b to be independent of temperature, is Rich- 

 ardson's well-known formula for the saturation-current from a hot 

 body as function of temperature. All of the multitudinous obser- 

 vations agree with it; but this docs not mean so much as might be 

 thought, for the experts inform us that all the data, no matter how 

 accurately taken, would agree quite as well with a formula in which 

 T, or T-, or even 7", stood in the place of the factor 7'' by which 

 the exponential is multiplied. Incidentally this would permit us to 

 make n vary as some small power of temperature, such as the inverse 

 square root, if we chose to make the resistance-temperature relation 

 in (4) agree with experiment at such a price. Or if we assume n 

 independent of temperature, we can calculate it from measurements 

 on thermionic saturation-currents. The measurements usualK' give 

 for n values of the order of magnitude of tin- number of atoms per unit 

 volume. 



What is more definitely signilicant is, that the \elocilies of the 

 emerging electrons are actually distributed in a manner compatible 

 with the assumptions made. Let us enquire iiow many of the electrons 

 issuing from unit area of the metal have velocities of which the x- 

 comi^onent lies between /( and u+dii. These are the ver>' same 

 electrons which struck the surface from within, having velocities of 

 which the .v-component lay between u' and it'+dit'; ii' and <lii' being 

 related to u and ii+dit by the equations: 



h)tir + eb = hn{ti'y-, ii'dii' = itdu. (9) 



'{"lie number of these electrons is by (()) 



dl =—-= e ZkTdu', (.10) 



-y'lirkT/m 



which by \irtue of the relations (it) reduces to 



e *r- — /> ikTdu, V") 



y/2irkTim 



which is identical with (ti) except for a constant factor; wliicli means 

 in turn that the distribution-function of the emerging electrons is 

 identical with the distribution-function of the internal electrons, 



