242 Mr. J. J. Thomson on (he Electric and Magnetic 



cular to the direction of motion of the sphere and to the mag- 

 netic induction ; and if co be the resultant velocity of the 

 sphere, and 6 the angle between the direction of motion of the 

 sphere and the direction of magnetic induction, the magnitude 

 of the force 



fjbe 



= 9o)N/a 2 + Z> 2 + c 2 sin6>. 



It -vA'ill be useful to endeavour to calculate the magnitude 

 of this force on a particle of air moving in a vacuum-tube ; 

 although our knowledge of the magnitude of several of the 

 quantities involved is so vague that our result must only be 

 looked upon as showing that the force is of an order great 

 enough to produce appreciable effects, and must not be looked 

 upon as having any quantitative value. 



Let us suppose that the mass of a molecule of air is 10~~ 22 

 (C.Gr.S. system) ; that a, the radius of the molecule, = 10~ 7 ; 

 that, as before, <? = K x 3 x 10 12 a 2 = K x 3 x 10 -2 (this quantity 

 is probably enormously underrated) ; and as we know nothing 

 about the velocity of the charged particles, let us assume it to 

 be the mean velocity of the air-molecules, viz. 4 x 10 -5 . We 

 shall suppose the vacuum-tube placed in a magnetic field 

 whose strength is 10 3 . Then, by the formula, the accelera- 

 tion of the particle of air when the magnetic force is at right 

 angles to its path is about 10 7 ; this acceleration would pro- 

 duce a deflection of about 2 millims. per decimetre of path, a 

 deflection which could easily be observed. We know from 

 the experiments of Mr. Crookes and others that a magnet pro- 

 duces very decided deflections of the molecular streams ; and 

 the direction of the deflections (see Phil. Trans. 1879, part 1, 

 pp. 154 & 156) agrees with that given by formulas (5), if we 

 suppose that the particles projected from the negative pole are 

 negatively charged. 



§ 6. Let us now calculate the expression given by Max- 

 well's theory for the force between two charged moving par- 

 ticles. 



Let u, v, w be the components of the velocity of the centre 

 of one of the particles, v! , v' ', w' those of the other ; let R denote 

 the distance between the particles, e the charge on one of the 

 particles, e f the charge on the other ; let r denote the distance 

 of a point from the centre of the first particle, / the distance 

 of the same point from the centre of the second particle. We 

 shall suppose, for the sake of simplicity, that the particles are 

 very small ; we shall calculate the kinetic energy of the system 

 and deduce the forces between the particles by means of 

 Lagrange's equations. 



