On Thermionics. 825 



this may be written 



oo r*. 



2zyofl + ^/l+2Z- e ^ , ) 



i—Zq \ m tv 2 / 



m tv - 



e~ v \lv 



a— z \ v m w 2 ' 



/»2V /C»l 



( a-z 



X \ e~ u ' du. 



4'S^°(l+ v/l+2Zl?=£o) 



To get the current from a hot strip of the plane A of infinite 



length, bounded by the lines x = % 2 and #=fi, which is 



received by unit length of a strip of B bounded by the lines 



x = x 2 and x = x x , we may integrate with respect to both 



I I 



y and y from + -^ to — ^, where Z is any very great length, 



and divide the result by L In this case we get 



i = j I ? rf«/o I d.r I — — w>o ^~ A '" IW '° 2 dw ) e ~ V ~ dv 



a— z V v jjiicq* / 



2 a— z Q \ v ??i w 2 / 



This, as it should be, is equal to the expression (4) on p. 894 

 of the author's paper on the " Kinetic Energy of the Ions 

 emitted by Hot Bodies " (Phil. Mag. [6] vol. xvi. 1908), 

 which was obtained by a slightly different method. 



If the area of B to which it is desired to find the current 

 is not rectangular, the limits of integration with respect to 

 cc and y will have to be suitably adjusted. For example, if 

 the area is bounded by the circle in the plane z = a, whose 

 equation is (x — «r 3 ) 2 + (y—yzf = ^ 2 , the limits for x will 

 be x z + ^h 2 —(y—y^f and for y, y z + b. 



If we take the negative signs in equations (11) we shall 

 obtain the points where the trajectories cut the plane B the 

 second time. In this case the value of % is 



X = i^r )2 1 1 - V i + w ) ■ 



The most interesting application of this is obtained when 

 a — z is made to approach the limit zero. The planes A and 



