198 Prof. Challis on the Hydrodynamical Theory of the 



there results for determining the form of the function F the 

 equation 



« d¥ d¥ ..p N n 



This partial differential equation integrated in the usual way 

 gives 



¥= ~Ei(z-h) 'H^A/ 



Now it is certain that the expression for the velocity F(R, z) 

 must involve the distance r of the point P from the axis of the 

 wire. This condition is satisfied by the above value of F by 

 assuming that 



R-«\ /_ (R-«) 2 \-* 





and can be satisfied in no other way. We have therefore the 

 unique solution 



f= 9i = 9i, 



C 2 being an arbitrary constant. Thus exact expressions for the 

 velocity in any plane transverse to the axis and for that parallel 

 to the axis having been found, the total motion, which is com- 

 posed of these two, is completely determined. 



39. From the above expression for F, that which applies to a 

 straight cylindrical wire may readily be deduced. For putting 

 a + a for ft, a being a variable quantity restricted within compa- 

 ratively small limits, and giving to C 2 the form d a, d being an 

 arbitrary factor, we have 



F = 



d 



which for a straight wire, for which a is infinite, becomes -, 



r 



agreeing with the result obtained in art. 35. 



40. It would seem that the foregoing investigation might be 



generalized so as to apply to a wire conductor of any form, when 



it is considered that the determinations in arts. 30-35 of the 



forms of F(r) and/(r) for a straight cylinder did not involve the 



length of the axis, and would remain the same for a cylinder of 



infinitesimal length if the condition of circular motion about the 



axis were satisfied. We have shown that this condition is in fact 



satisfied by a uniform conductor of circular form, which may be 



regarded as made up of a series of right cylinders of infinitesimal 





