144 Dr. Barton and Mr. Morton on the Criterion for 



that the resulting quadratic has imaginary roots. Now if 

 account be taken of the distribution of the current in the 

 wire, the well-known work of Maxwell and Lord Rayleigh* 

 shows that the above differential equation must, for a straight 

 wire, be supplemented by terms on the right, 



~^R«V^+A^V^-rloR"V^&c., . . (2) 



where a is the length of the wire divided by its resistance 



I p I, and 11 the permeability of the material of the wire. 



For a curved wire we should probably have an equation of 

 the same form with different coefficients. The question is, 

 how do these additional terms affect the condition for oscil- 

 latory discharge ? 



The added terms have coefficients which are in general 

 small compared with R. Lord Rayleigh (loc. cit.) takes, for 

 iron, the value 10 4 for resistivity and 300 for //,. This would 



give for a wire of radius a, /*«= nA ; so that the coefficient 



d 3 Q 

 of -j~ would be, even for thick wire, less than RxlO -5 . 



For copper the value would be still less. If, therefore, we 

 put e xt for Q, we have an algebraic equation of which the 

 terms above the second are of small and decreasing importance. 

 The effect of these small terms will be (1) to introduce very 

 large roots corresponding to very rapid oscillations ; these 

 will clearly be of small amplitude, and will not affect the 

 main phenomena of discharge: (2) to modify the original 

 roots of the uncorrected quadratic equation. The cases of 

 oscillatory and non -oscillatory discharge are separated by the 

 case of equality of these displaced roots. 



It is easy to see that the effect of the added terms will be 



to make the critical value on the simple theory ( = py ) 



correspond actually to an oscillatory discharge. For in this 



case the graph of y=. ^ •+ R# + La? 2 evidently touches the axis 

 \j 



of a? at a point on the negative side of the origin, viz. 

 ■p 



x= — ^y-, and lies entirely above the axis. If we compound 

 with this the graph of the additional terms, 

 y=— ^RaV^ 3 &c, 



* Maxwell, ' Treatise,' vol. ii. § 690 ; Rayleigh, Phil. Mag. vol. xxi. 

 p. 381 (1886). 



