Theory of Magnetism. 47 



constant /u,. Then we shall have, at any point exterior to the 

 circle of radius y, 



U=^cos0, W=-£,sin<9. 



Consequently, at points for which = and 0=7r, W = and 



XJ = %■ reckoned in the direction from A towards B; and at 



points in the plane through C transverse to the axis, U = and 



W= — 77-' Hence at the same distance r, the backward motion 



across that plane parallel to the axis is half the forward motion 

 along the axis ; and each of these velocities varies as the cube of 

 the distance from C. 



Since y is an unknown disposable quantity, the above suppo- 

 sition that VR 3 , or V(?/ 2 + ^ 2 ) ¥ , is equal to a constant, is not ille- 

 gitimate. The function that y is of x will depend on the form 

 of the magnet. In the case of a cylindrical magnet y will not 

 generally differ much from the radius. It is also to be remarked 

 that the above value of U for a point on the axis, and that of W 

 for a point in the transverse plane, are to be considered as ap- 

 proximative functions of r. The more complete values would pro- 

 bably be of the form 



The motion in these magnetic streams is an instance of steady 

 motion for which udx + vdy + wdz may be assumed to be an 

 exact differential. This may be maintained on the principle that, 

 after the impulse is given to the fluid within the magnet in the 

 direction of its axis, the consequent curved courses of the lines 

 of motion are determined solely by the mutual action of the parts 

 of the fluid. Also there may be reason to conclude that for fluid 

 of unlimited extent that expression is an exact differential in any 

 case in which the lines of motion may be cut by surfaces of con- 

 tinuous curvature — that is, whenever the motion is proper to a 

 fluid, and not such as a fluid is capable of when it may be con- 

 ceived to consist of parts that are solid. Leaving, however, this 

 point for future consideration, I shall now assume, for the rea- 

 son given above, that udx + vdy -f wdz is an exact differential 

 for magnetic streams. In that case, as is known, the relation 

 between the density p 1 and velocity Y x for the streams of a given 

 magnet is expressed by the equation 



Pi = Po e 2a>, 



