On Tliermionies. 819 



portion of i/r = which is cut off by some closed curve 

 the limits for u v will be determined partly by the bounding 

 curve and partly by the locus of the tangents. With this 

 understanding as to the meaning of the double integral we 



n o o 



shall therefore have 



i= n 1 j dS l dWfX \ f(w ) fi(u ) f 2 (v ) du dv . (10) 



as an equivalent of the expression previously found. It 

 will often be possible so to choose the direction of ic that 

 f(ivo) does not depend on u and v Q . 



§ 4. The Initial Boundary Conditions, 



Each element dS of the surface A is at any instant 

 shooting off a large number of ions in different directions 

 and with all possible speeds. The researches of Richardson 

 and Brown * and of the author f have shown that the number 

 of ions leaving the element of surface d$ in unit time which 

 have values of any velocity component v within a given 

 range, say between u and ii + du , is approximately identical 

 with the number given by Maxwell's Law of Distribution 

 for the number of molecules of a gas having the same mole- 

 cular weight crossing an equal and similar surface in equal 

 time with the range of the velocity component u within the 

 same limits, the temperature of the gas being the same a* 

 the hot metal, and its pressure equal to the equilibrium pres- 

 sure of the ions. It may be permissible to say here that 

 more elaborate experiments now in progress have only 

 tended to confirm this conclusion, and, so far as they have 

 gone at present, point to the law being an exact one. It is 

 to be remembered that we are dealing here, not with the 

 distribution of velocity among the different molecules in a 

 given volume, but with the distribution among those leaving 

 a given surface in a given time, which is a different thing. 



Assuming Maxwell's law to hold exactly we can write 

 down the functions such as /i(uq) /sOto) and f s (w ) which 

 determine the initial frequency of a velocity component 

 within a given range. They will depend both on the kind 

 of axes chosen and on their orientation relative to the surface. 

 The following list, which embraces all the more important, 

 cases, may easily be verified. In each case n is the total 

 number of ions emitted per unit area in the interval 

 considered. 



* Phil. Mag. [0] vol. xvi. p. 353 (1908). 

 t Phil. Mag. [6] vol. xvi. p. 890 (1908). 



