50 Prof. Thomson, On the motion of a 



e the charge on the particle. The components of the force on the 

 particle are 



Xe 4- e (fiw — <yv), 



Ye + e (<yu — aw), 



Ze + e (av — f3u). 



Hence, since the velocity of the particle is proportional to the 

 force we have, k'bemg a coefficient depending on the nature and 

 pressure of the gas, 



eku = Xe + e (/3w — <yv), 



or Jcu + yv — (3w = X. 



Similarly, 



— yu + kv + aw = T, 



/3u — av + kw = Z. 



Solving these equations we have 



k*X + Jc(yY-/3Z) + a(«X + £7 + yZ) 



u = 



v = 



w = 



k 3 + k (a 2 + yS 2 + 7 2 ) 



k*Y +k(aZ- y X) + ft (aX +/3Y + yZ) 

 k 3 + k (a 2 + /3 2 + y 2 ) 



k*Z + k (j3X - aY) + 7 (aX + /37+ yZ) 

 k 3 + k (a 2 + ft 2 + 7 2 ) 



We see at once from these equations that when the magnetic 

 force is small compared with k, u, v, w are proportional to X, Y, Z, 

 that is, the particle follows a line of electric force ; on the other 

 hand when the magnetic force is large compared with k, u, v, w 

 are proportional to a, /3, 7, that is, the particle follows a line of 

 magnetic force : thus in very intense fields the paths of these 

 particles will be the lines of magnetic force. In the general case, 

 if H is the magnetic and F the electric force, and 6 the angle 

 between them, the velocity has a component along the line of 

 electric force proportional to k 2 F; another component along the 

 line of magnetic force proportional to H 2 F cos 6, and a third 

 component, at right angles both to the magnetic and electric 

 forces and proportional to kHF sin 6. In this case the path is 

 neither along the line of electric nor magnetic force, but is a 

 spiral. 



When the electric force is radial and the magnetic field constant 

 we can find the equation to the path of the particle. Take the 



