142 Mr. G. F. 0. Searle on the 



obtain some interesting results when n or u/v is so small that ? 

 as we see by (1), the eleetric force due to the system may be 

 regarded as being the same as if the system were at rest. 

 The axis of x will be taken in the direction of motion, the 

 volume integral of Fj 2 through all space will be denoted by 

 2E 2 , and 7) will be written for K/47n; 2 , or |uK 2 /47r. 



If M 1? M 2 , M 3 be the components of the mornentuni._when. 

 the axis of the system is parallel to the axis of x, we have, by 

 (24), since fjuKv 2 — ! 



M l ==^S(E/ + E 3 2 )=^2(E 2 -E 1 2 ) = 2(W^ 2 )U -w7 7 2E 1 ^ 



while, by (26), w T e have, for reasons of symmetry, 



M 2 =-«172E 1 E 2 = J M 3 =-^SE 1 E 3 = 0. . (41) 



In the expression for M 2 we have put U for U, since we 

 know, by § 18, that, when n is small, U — U is proportional 

 to n 4 . Since there is symmetry round the axis_, 2E 2 2 = 2E 3 2 > 

 and thus 



w n %E 1 2 = 2iiTJ /v"-M 1 , ur ] %F^ = u V ZF 3 2 = ±M. 1 . . (42) 



Now let the system be turned round the axis of y in the 

 positive direction, so that the axis of the system makes an 

 angle 6 with the direction of motion and with the axis of «r, 

 and let G be a point rigidly connected with the system and 

 turning with it. Since when u/v is very small the electric 

 field turns round with the system without any other change, 

 the new components of the eiectric force at G are given by 

 the equations 



Ei'^EjCosfl+Eosin^ E/ = E 2 ) 



>. . . (43) 



Es'^-EiSintf + E 3 cos0 ) 



For M/, the component of the momentum in the direction 

 of motion, we have, as in (40), 



M 1 ' = 2uU /v 2 -u7 ] 2 l Fi 1 /2 . 



But SEiEg^O, and hence, by (43), 



Mi' = 2uTJol v 2 - urjt{ E : 2 cos 2 6 + E 3 2 sin 2 6) , 



= 2mU sin 2 0/v 2 + Mj (cos 2 6 - J sin 2 6) . 



If N be the momentum in the direction of motion when the 

 system moves at right angles to its axis, i.e., when 6 — } 2 tt, 

 we have 



N^Uo/^-pii, (44) 



and thus for any value of 0, 



M/=M 1 cos 2 6' + Nsin 2 6'. 



