PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
283 
A 3 . 
87r 2 p 2 cro- , d 3 a /3 
d 2 d x d ' lP y 2 
sin 2</>, 
where A cos pt is the intensity of the magnet and terms of the eighth degree in the 
radii are omitted. 
Thus, to this order of approximation, we have the following results :— 
(a.) The couples exerted are independent of a, i.e., of the direction of the axis of 
the electromagnet. 
(f3 .) The shells have equal couples in opposite directions, the parts of the shells 
directed towards one another being attracted towards the electromagnet if </> is less 
than a right angle, and driven away if <f> exceeds a right angle. 
Fig. 5. 
Fig. 6. 
If (f> be a right angle the couple is zero. 
We next discuss the case of a thin spherical shell in any field and show that— 
(a.) If the field be symmetrical round any diameter, there will be no couple about 
that diameter. 
(f3 .) If the external field be completely in the same phase, or if, when the external 
magnetic potential is expanded in harmonics over the surface, the terms of the 
harmonic of any degree are in phases that are the same for the same degree, then the 
couple will vanish. From this it follows that if the primary field be in the same 
phase throughout, and any number of perfectly conducting bodies be introduced, 
their currents will be in the same phase as the primary field, and no couple will 
be produced. 
It has been stated that in the case of Professor Elihtt Thomson’s sphere, spinning 
on a sheet of copper, the effect was due to the sheet acting as a “ shield ” and pro¬ 
ducing an uusymmetrical fiekl. That this explanation is not satisfactory will be 
evident on considering that the field is unsymmetrical before the sheet is interposed, 
and that the better the sheet conducts the better the shielding effect; so that if it 
be a perfect conductor a large couple would be expected whereas in reality there is 
none whatever. 
The effect must, I think, be traced to the fact that the currents induced in the 
sheet are caused by self-induction to lag, so that the field in action on the sphere 
does not alternate in one phase. 
2 o 2 
