196 ADVANCED ELECTRICITY AND MAGNETISM. 



and the general solution of this equation is : 



q = Qr* sin (/ - 6) (5) 



where Q and 6 are constants of integration, and a and co are : 



R 



"=^L ^ 



and 



R* 



Derivation of equation (4). Let q be the charge on the 

 condenser C in Fig. 133 of Art. 112 at a given instant. Then 



dq , d?q 



is the current m the circuit, and is the rate of increase 



at dr 



of the current. The electromotive force which acts on the circuit 

 is used (a) in part to hold the charge q on the condenser (the 



part so used is equal to J , (b) in part to overcome the resistance 



of the circuit (the part so used is equal to Ri or R 1 , and 

 (c) in part to make the current increase (the part so used is equal 



to L or L \. But when the system shown in Fig. 133 is 



oscillating freely there is no electromotive force acting upon it 

 from outside and therefore the sum of the three parts (a), (b) 

 and (c) is zero. 



THE DIFFERENTIAL EQUATION OF WAVE MOTION. 



114. The equation of a traveling curve. Let cc, Fig. 135, be 

 a curve which is stationary with respect to the origin 0', the 

 equation of the curve with respect to the origin 0' is: 



y = F(x') (i) 



Suppose, however, that the curve cc and the origin 0' are both 

 traveling to the right at velocity v so that the abscissa of the 

 moving origin 0' with respect to the stationary origin is vt 

 as shown. Then the abscissa x of any point on the curve cc 



