Motion of an Electrified Sphere. 47 



When u x and u 2 are parallel but in ojjposite directions, so 

 that cos a= — 1, 



When the motion is exactly reversed, so that cos ci— — 1 

 and n 2 = n x 



W^Ii^log^-2} (13) 



The first result (11) was given by Dr. Oliver Heaviside*. 

 These results have also been deduced by Paul Hertz f on 

 dynamical principles from the values of the energy and 

 momentum o£ a charged sphere in steady rectilinear motion. 



§ 5. We now pass on to the general case in which the 

 initial and final velocities are inclined at any angle. The 

 integration of F(w 1? n 2 , a) is now somewhat complicated, but 

 on account of the fundamental importance of the integral 

 two independent methods of evaluation are given below. 



§ 6. The first method of evaluating F(/i 1 , n 2 , a) is due to 

 Mr. Gr. T. Bennett. Since this method is partly geometrical, 

 it will be convenient to modify the integral so as to exhibit 

 the geometrical quantities involved. We see, at once, from (7) 

 that 



/R R \ M 2 T n£s 



f &d<o m . 



] (^ 1 -Rcos^ 1 )(A 2 -Rcos6' 2 ) , * * {lD) 



where T i* R 2 d 



Here 1 and 9 2 are the angles between the radius defined 

 by dco and two fixed radii drawn from the centre of a sphere 

 of radius R, and thus we can write 



_ rwda 



J 



(16) 



where 2 ls r 2 are the perpendiculars drawn from any point on 

 the surface of the sphere upon two fixed planes which cut 

 the two fixed radii at right angles at distances hy and h 2 from 

 the centre of the sphere. Since n r <i and /? 2 <1 we suppose 



r Unterwclmngen iiber umtetige Beweyimgen eines Electrons. Got- 

 tingen, 1904. See also M. Abraham, Theorie der Blectriz/'tat, vol ii. 

 p. 233. 



