the Oscillatory Discharge of a Condenser. 145 



which, for negative x, also lies above the axis, we get the 

 parabolic graph displaced upwards, so that the roots of the 

 modified equation become imaginary. 



To find to any desired degree of approximation the condition 

 for equality of the chief roots of the equation, we may use the 

 principle that a repeated root of an equation f(x) =0 is a root 

 also of the derived equation f (x) = 0. 



In what follows we shall write the differential equation of 

 discharge in the form 



and the corresponding algebraic equation 



0= ^+Rx + Lx 2 + <f)(x), .... (4) 



using (p for the series of small terms in a 2 ^ 2 , &c. 

 The derived equation is 



= R + 2J,x + (j) / (x). ..... (5) 



Let the common root of these equations be x= — -~j- + 0, 



where is small; then we can find to any degree of 

 approximation from (5), which becomes 



= 2L0 + ( / + <fc,".0 + i£o'".0 s + Ac.,. . . (6) 

 w here <f) is put for </> ( — ,jy- 

 This gives 



Equation (4) becomes, in terms of 0, 



Puttiug in this the value (7). we get the required condition 

 in the form 



r!~4L ^ 0+ 4L «L 2 ' * * ' ' W 



which goes to the third order in <£ or the sixth in *//.. 

 Phil. Mag. 8. 5. Vol. 48. No. 290. July 1899. L 



