PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
313 
[It might be objected that being given the external field, we have no right to take 
F 0 = 0 at the surface ; we have proved, however, in our introductory work, that 
if 12 0 be given, an alteration in the normal component of the vector potential only 
introduces an alteration in the electrostatic potential.] 
18. Hence it follows that— 
(a.) If the external field be symmetrical round the axis of Z, so that 12 is 
independent of <f>, the couple will be zero ; 
(l3 .) if the external field be completely in the same phase, or if when expanded in 
harmonics over the surface its form is 
n Q = t U« COS ( pt + €*), 
where e„ is independent of the coordinates 0, <£, then the couple vanishes. 
From this it follows that if we have an exciting field in the same phase, and intro¬ 
duce any number of perfectly conducting bodies, the couple will still be zero. 
The Couples on two Thin Spherical Shells due to the presence of an Alternating 
Current in a Straight Infinite Wire. 
19. Let the line joining the centres of the shells he taken as OX and the shortest 
distance between this line and the infinite wire as OY. 
Let the radii of the shells be a, a', and their central distances from O, h and h ': 
let the distance from O of the wire be c. 
Fig. 35, 
If a point in space have polar coordinates r, B, </> when referred to Q as origin and 
axes parallel to OX, OY, OZ, and coordinates /, <f>' to Q' as origin, its Cartesian 
coordinates will he 
2 s 
MDCCCXCIT -A, 
