540 Mr. C. G. Darwin on the 



work only in cases where the velocities are and remain fairly 

 small fractions of the velocity of light. In this way the spiral 

 orbits will be excluded and the approximation to terms in 

 C -2 will represent the facts as closely as is required. This 

 limits our method to the problems of spectroscopy, and cuts 

 out such interesting questions as the collisions of high-speed 

 (3 particles. 



3. The variability of mass of an electron is usually deduced 

 from considerations of electromagnetic momentum. It is 



in v 

 found that the linear momentum is — , where m is the mass 



P 

 for low velocities, v is the constant velocity, and 



^Vi'-i) (1) 



By well-known arguments (which, however, cannot quite 

 escape criticism) there follow the equations of motion of the 

 type 



where F x is the total force on the particle in the x direction. 

 The three equations of the type (2) form our starting-point. 



The variability of mass is often expressed by considering 

 the quantities m//3 and m//3 3 as transverse and longitudinal 

 mass respectively, but these expressions are in fact deduced 

 from (2), and it is useless to put down the more complicated 

 equations in terms of them and then retrace the steps of the 

 argument back to (2). As long as the equations of motion 

 are expressed in terms of rate of change of momentum, 

 instead of mass acceleration, there is no need for the con- 

 ception of longitudinal mass. 



First consider the problem of a single particle of charge e 1 

 and mass m x in any field of electric and magnetic force, 

 variable in time and place. Making use of the vector 

 notation, let ^(='^7, y, z) be the position of the particle. 

 Let E and H be the electric and magnetic forces, </> and A 

 the scalar and vector potentials from which they are derived. 

 Then 



E=-A<£-£^ andH = curlA, . . (3) 



and (2) becomes 



& ( m i - 1 a jl ^idA , e lr . . ._ ... 



where /3 i = >/(l— *i 2 /C 2 ) in the same notation. 



