196 Prof. Challis on the Hydrodynamical Theory of the 



the current in opposition to the tendency of the inertia of the 

 fluid to put a stop to it. In the present theory this stress is due 

 to the action of the battery ; the wire supplies a channel for the 

 current; and, as is shown in art. 11, it is dynamically necessary 

 that the current should flow in a complete circuit. 



35. It is clearly possible that the form of the function F(r) 

 might be determined by arbitrary conditions. For instance, if 

 the above-mentioned stress were arbitrarily caused to be the same 

 at all points of the transverse plane, the velocity parallel to the 

 axis would be the same at all points, and F(r) would be a constant. 

 But it is evident that this is not true of a galvanic current. The 

 principle of the present inquiry demands that as a definite rela- 

 tion between the functions f(r) and F(r) was obtained in a unique 

 manner by integration, the form of the function F(r) should be 

 similarly determined. Now the only way in which that form can 

 be obtained exclusively by integration is to equate to zero the 



F(r) d.~F(r) 

 above factor — — H * , in which case integration gives 



dr 



F(r) = 



r 



Thus the velocity parallel to the axis varies inversely as the dis- 

 tance from the axis ; and the stress which maintains that velocity, 

 and is therefore proportional to it, varies according to the same 

 law. Since the transverse sections of the elementary channels 

 above defined vary directly as the distances, it follows that 

 through each elementary channel outside the wire the same 

 quantity of fluid flows in a given interval. Also, since it has 



c 

 been shown (art. 31) that /(/•) = - F(r), we obtain 



or the transverse circular motion varies inversely as the square 

 of the distance. These results are essential to the hydrodyna- 

 mical theory of galvanism. 



36. But for the theory of the action of a galvanic coil we re- 

 quire to know the motion of an setherial current along a fine 

 wire the axis of which has the form of a circle, and the trans- 

 verse section of which is circular and uniform. For this case it 

 will be assumed that, by reason of symmetry, the motion at any 

 given point is compounded of motion parallel to the axis, and of 

 motion in the plane passing through the point and the centre of 

 the axis, and cutting the axis at right angles. Let the plane of 

 this axis be parallel to that of xy, and its centre be on the axis 

 of z\ and let h be the height above the plane xy of the point of 

 intersection of the circular axis by the above-mentioned trans- 



