Fundamental Concepts of Electrical Theory. 411 



If the Newtonian equations of motion were not known, 

 mathematicians might perhaps endeavour to formulate a 

 scheme of equations of motion by using the principle that 

 two particles cannot occupy the same space at the same 

 time. In the present case the principle may be supposed to 

 be violated if the electrified particle cuts through a tube of 

 electric force of the given field, or if it leaves a tube of force 

 to which it has been attached. 



Let us consider the consequences of assuming that the 

 velocity u of the electrified particle is determined by the 

 equations 



I mu = A, \mc 2 = <S> (9) 



We should then have 



2 Ox ox c ot c ot 



Now it follows from the identities 



B(X,Y) _1 5(V,T) B(X,Y) 3(V,T) 



that both X and Y satisfy the equations 



da? a* + oy oy oz oz c 2 t ot ' 

 SX dT + c>X dT + dXdT = 1 3X BT 



0~x "dx ~dy ~dy oz ~dz c 2 ot ot ' 



Hence it follows that 



oX^ oX^ BX^ BX 



O^ O^' 0# O^ 



and that Y satisfies a similar equation. Consequently 

 X and Y remain constant during the motion of the elec- 

 trified particle ; and this means that the particle remains on 

 the same line of electric force during the ivhole of its motion, 

 it never cuts through a tube of force *. 



The law of motion embodied in equations (9) is so simple 

 that it is hardly likely to be correct, nevertheless we are 

 justified in retaining our hypothesis as a possible one until 

 it has been shown conclusively that it is contradicted by 



* In order to ensure that this may be true it is clearly only necessary 

 to assume that the velocity u is represented by a point on the line L. 

 Our assumption (9) is only one way of satisfying thi3 condition. 



