Action of a Galvanic Coil on an external small Magnet. 187 



X) — X o 



or, since cos 6 — - , sin 0=.-, and r 2 = (jo— a?) 2 + § 2 , 



x= 3 (^-^) 9 + j^ 2 Y= 3/*« 8 g (;?--#) 



, (( i? _ ( ^ + ^)f 2((p-*) 2 + ^)l 



Hence, to calculate the total velocity at P in the longitudinal 

 and transverse directions, we have to add the velocities due to 

 all the slices of given thickness dx from end to end of the mag- 

 net, or to obtain the integrals k\ ILdx and k\Ydx from x= —I 

 to x= -f-Z, the length of the magnet being 21, and k a constant 

 factor. The results will be found to be 



Longitudinal 1 „ ^f 3 f 2—1 P + l 1 



velocity J 2 t((j»-J)*+S*)* (> + /) 2 + 2 2 )* J ' 



Transverse 1 _ V^ 3 / 2 9 \ 



velocity / 2 L(^+ (/>-/) 2 )f (?*+(* + *)*)* J" 



16. It will now be shown that these velocities are propor- 

 tional to the directive actions of the magnet in the longitudinal 

 and transverse directions on a small needle having its centre at 

 P, and movable about an axis perpendicular to the plane con- 

 taining P and the axis of the cylinder. The small magnet will 

 be supposed to be surrounded by magnetic streams exactly like 

 those which, according to the foregoing theory, belong to the 

 large magnet, and to be of such small dimensions that the 

 streams from the large magnet may be considered to have the 

 same direction and velocity at all the positions of the atoms of the 

 other. To find the action of the large magnet on the small one, 

 it is now required to determine for any point the accelerative 

 -action of the pressure of the fluid resulting from the coexistence 

 of the two sets of motion. 



17. It is clearly not necessary to take account of any force 

 acting perpendicularly to the plane passing through P and the 

 axis of the cylinder, because all such forces are equal and oppo- 

 site on the two sides of the plane. Let, therefore, that plane 

 contain the axes of x and y, and let u v v } be velocities, parallel 

 to the axes, due to the large magnet, and u 2 , %\ be those due to 

 the small one. Then by hydrodynamics, the motion being 

 steady, and, as vanishing at an infinite distance, such as makes 

 udx + vdy + ivdz an exact differential, we have 



p=c-i((M 1 +u 8 ) sf +K+t» 8 5*). 



Let Vj be the velocity of the incident stream of the large mag- 

 net, and let its direction make an angle 6^ with the axis of x. 

 Then w l = V 1 cos 6 Y and v 1 =-Y l sin 6 V Again, let the velocity 



