TO THE THEORY OF MAGNETISM. 107 



will, in the first place, suppose a cylindric wire whose radius is a 

 and length 2X, is exposed to the action of a constant force, equal 

 to f, and directed parallel to the axis of the wire, and then 

 endeavour to determine the magnetic state which will thus be 

 induced in it. For this, let r be a perpendicular falling from 

 a point p within the wire upon its axis, and x, the distance of 

 the foot of this perpendicular from the middle of the axis; then 

 f being directed along x positive, we shall have for the value of 

 the potential function due to the exterior forces 



and the equations (V), (c) (art. 15) become, by omitting the su- 

 perfluous constant, 



(r), the distance^', do- being inclosed in a parenthesis to prevent 

 ambiguity, and j/ being the point to which ^r' belongs. By the 

 same article we have = S<f> and = 8^r, and as cf> and ty evi- 

 dently depend on x and r only, these equations being written at 

 length are 



Since r is always very small compared with the length of the 

 wire, we may expand </> in an ascending series of the powers of /*, 

 and thus 



X, Xv Xv etc. being functions of x only. By substituting this 

 value in the equation just given, and comparing the coefficients 

 of like powers r, we obtain 



*-z-^^+^ *' 



9 ~ dx* 2 a+ dx* 2 2 .4 2 + etC - 



