PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
317 
21. There are, of course, six similar equations obtained from consideration of the 
currents in the a , b' shell: these equations are accurate, since the introduction of 
further approximations gives rise to harmonics of higher orders than the first. 
From the four equations giving C, F, C' and F', it follows that all these quan¬ 
tities are accurately zero, as might have been expected, since the system is unaltered 
on taking — 2 for -f- 2 , i.e., putting (tt — 9) for 6. 
From these equations it follows, as in the case of the cylinder, that the principal 
terms of the first harmonics are of the first degree in the radii, but their next are of 
the fourth (not the third, as for the cylinder). 
It will also be obvious that the principal terms of the harmonics of the second 
order will be of the second degree in the radii. 
The values of the coefficients can be calculated with ease, as with the cylinder, but 
we wait to see which of them are involved in the couple. 
Writing the external field on the a, b shell in the form 
n 0 = Y : cos pt + Z l sin pt, 
the couple will be (xii.) 
V 3Z, 7 SY, 
. 1 0^. 1 '4 j 
and the most important term omitted (that from the second harmonic) is of at least 
two degrees higher in powers of the radii. 
Also, 
Y 1 = 'A sin 9 cos (f> + ' B sin 9 sin <f>, 
Z 1 = 'D sin 9 cos (f> + 'E sin 9 sin <f>, 
so that the couple is 
9 pa 
4A 
- || dS [('A cos (f) -\- 'B sin <£)(—'D sin <£ + 'E cos <fi) 
— (— 'A sin (j) + 'B cos <£) ('D cos <f> + 'E sin <£)] sin 2 9 , 
or 
or 
9 pa 
4A 
- f 89. a 2 sin 3 9 T 8<f> {('A'E - 'B'D) (sin 3 <f> + cos 3 <j>)}, 
i Jo Jo 
9 pa- 
4A 
.fa 3 . 2rr ('A'E — 'B'D).(xiv.). 
On reference to the values of'A, &c., it will be seen that to fourth powers of radii, 
'A'E — 'B'D = - ~ [cF/ + 26 D '] aa ' 
r 
'2 
■ (f ’ 
also 
