78 APPLICATION OF THE PRECEDING RESULTS 



condition, combined with the equation (b), will completely deter- 

 mine the value of DV, and we shall thus have the value of the 

 potential function V+ D V, at the instant of time t + dt, when its 

 value F, at the time t, is known. 



As an application of this general solution ; suppose the body 

 A is a solid of revolution, whose axis is that of the co-ordinate 

 z, and let the two other axes X, Y, situate in its equator, be 

 fixed in space. If now the exterior electric forces are such that 

 they may be reduced to two, one equal to c, acting parallel to 

 z, the other equal to b, directed parallel to a line in the plane 

 (xy), making the variable angle (f> with X; the value of the 

 potential function arising from the exterior forces, will be 



zc xb cos <j)yb sin <f> ; 



where b and c are constant quantities, and <f> varies with the 

 time so as to be constantly increasing. When the time is equal 

 to t, suppose the value of V to be 



V /3 (x cos r + y sin w) : 



then the system of lines L, L', L" will make the angle w with 

 the plane (xz), and be perpendicular to another plane whose 



equation is 



= x cos r + y sin r. 



If during the instant of time dt, <f> becomes < 4- D<j>, the aug- 

 mentation of the potential function due to the elementary change 

 in the exterior forces, will be 



jy 7= (x sin (j> - y cos <) bD<j> ; 



moreover the equation (b) becomes 



dDV , . dDV ,, 



= cos -sr . -y F- sin vf . 7 [01 



dx dy 



and therefore the general value of D V is 



D F= DF (y cos r x sin ;); 



jD^ 7 being the characteristic of an infinitely small arbitrary func- 

 tion. But, it has been before remarked that the value of D V 



