Tlieory of Metallic Conduction. 557 



whence if 

 we see that 



whilst 



uf - %t 2 + Vt 2 + ft 



^ + 2 1 (Xf^-f-Y^ + ZftJ^ 



■r 



f tl==i 



r 



with similar expressions for y^ and Zt : (^'2/ -2) are of course 

 the coordinates of the point from which the electron started. 

 We shall write these expressions in the form 



u t 2 =u 2 + i u 



x t = x + ix, y t =y+ i y , - t = ~ + iz, 



denoting the integrals for the increments of the respective 

 quantities by i. 



But now we may interpret the number of the electrons 

 under consideration (SN) in terms of their position and 

 velocity coordinates at the time t, so that 



8N« = 8N = F (*/* 2 — /„, x t — i x , yt—iy, z t -iz, t—r) e~ ^ ■' - dV. 



It is now a fundamental assumption in this theory that 

 the effect of the accelerations produced by the external fields 

 is so small that the increments i of the various quantities 

 concerned are all so very small compared with these quantities 

 themselves, that a sufficient approximation is obtained by 

 neglecting second and higher degree terms in these quantities. 

 We may therefore write the above expression for SN« in the 

 form 



v o(w) ox oy ' o~ at / r m 



wherein F 0f =F (w* 2 , x t , y u z t , /), dV t = dtj t dr\ t d% u 



If now we keep (f t iVb fts***^^ constant, and inte- 

 grate this expression for SN« over all values of t, we shall 

 obtain the complete group of electrons which per unit 

 volume round the point (#*, i/t, z t ) have velocities between 

 (fc Vu ft) and [tjt + dZh Vt + dt]t> ft + e/ft) at the time t. Thus, 

 dropping the suffix /, we have 



^— F — 1 \lu vr-.; V -f > z ^— + l y -<— + I ST" + T 3— J* 



* 



