rapidly moving Electrified Particles. 469 



layers of different substances will produce the same deflection if 



their thicknesses are proportional to Xjff^ ; since X = -^ ,^ we see 



that this implies that the results expressed by equations B are 

 true even when the deflections are not small. 



In the preceding investigation we have supposed that the 

 angular deflections were all in one plane, the differential equation 

 satisfied hy f{z,<l>) when the deflections take place in any plane 

 may be found as follows : 



As before the particles for which z = z ■\-\cos^, and (f>=4>i 

 must have come from the particles determined by z and (f)o where 



cos ^1 = cos ^2 cos 6 + sin ^2 sin 6 cos'^jr, (1) 



i/r is the angle which the plane in which the deflection 

 takes place makes with the plane through the original direction 

 of the particle and its direction just before it experiences the 

 deflection 6. As all directions of yjr are equally probable the 

 probability of -yjr being between -vlr and yjr + dy^r is dy^jzir. 

 Hence, 



f{z + \ cos 0, <^i) =j-^f(z, (f),), 



by Taylor's Theorem, the right-hand side is equal to 



+ i;|^W„"^(*-^'^' (2). 



From equation (1) we get 



4>2 — <f>i— ^ cos ^jr — ^d^ cot (pi sin^ \/r, 

 substituting this value of ^2 — <f>i in equation (2) we get 



Xcos^/^=-i^^cot^^^+-^, 



4Xd/_ 1_^ 1 d-^f 



0^ dz sin <f) d<f) cos (f) dcj)^ ' 



or if cos (}> = t, the equation may be written 



^df^l-^'d^ 

 d' dz~ t dt^' 

 or with the same notation as before, 



^ df ^l-t^ d?f 

 dz t df ' 



