PEIRCE. 



MANNER OE GROWTH OF A CURRENT. 



513 



where either a or n may be assumed at pleasure and the other computed 

 by means of the equation 



prc 2 = 4tt p a 2 . (13) 



Since the core of the solenoid is solid, Bessel's Functions of the second 

 kind will not be needed in the problems of this paper, and we may assume 

 that H is expressible in an infinite series of terms of the form 



J (nr). 



(14) 



If, after the current in the coil has been for some time steady and 

 the core has become uniformly magnetized, the coil circuit be suddenly 

 broken so that the duration of the spark is less than a thousandth of a 

 second, the intensity of the magnetic field at the boundary of the core 

 where r= a falls suddenly to, and remains thereafter at, zero, and the 

 normal form (14) will satisfy the condition H a = 0, if such a value be 

 chosen for n as shall make na a. root of the equation 



J (x)=0. (15) 



A sufficient number of these roots for the purposes of this paper can be 

 found 6 in almost any book on Bessel's Functions, with the correspond- 

 ing values of J x (x) : the first twelve are given in Table I. 



The pth root in order of magnitude of the equation Ji(x) = is 

 denoted by x p . 



If the intensity of the uniform magnetic field in the core before the 

 break was H Q , we have 7 at any time (f) after the break and at any dis- 

 tance (r) from the axis of the core, 



6 Byerly, Treatise on Fourier's Series and Spherical, Cylindrical, and Ellip- 

 soidal Harmonics, p. 286 ; Gray and Mathews, Treatise on Bessel's Functions, 

 p. 244 ; Peirce and Willson, The first 65 roots of the Equation J (x) = 0, The Bulle- 

 tin of the American Mathematical Society, 1897. 



7 Byerly, Treatise on Fourier's Series, etc., p. 229 ; Heaviside, Electrical Papers, 

 1, 391. 



VOL. XLI. — 33 



