PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
299 
(• 2ir 
In finding the mean value of the couple, a 2 | apv 80, we bear in mind that the 
mean values of sin 3 pt and of cos " pt are and of siny>£ . cos pt, zero. On picking out 
the coefficients of sin 2 pt and cos 3 pt in a 0 w, we find for the mean value of the couple 
(— M # sinn0 + N„cosn0)(27rfflp{M»cosn0 + N„sinn0} 
— cm{Q«cosn0 + R„sinw0j) 
+ ( — Q„sin nO -f R„cosw0)(<xn{M„cos?i0 + N„sin nO} 
+ 27rap{Q „cosn0 + R„sinn0}). 
Also 
f sin 2 n6S6 = ( cos 2 nO $9 = n, 
Jo Jo 
and 
2t r 
sin nO cos nO SO = 0. 
0 
f £0 i _ ~ an l } 
I o n=i 2 (4tt 2 « 2 p 2 + cr 2 ?r) 
Thus the mean value of the couple becomes 
iravpp 
! 2 (47r 2 a 2 p 2 + cr 2 w 2 ) 
27 Tap{— M„N„ + M„N„ — Q„R* + Q«R;J 
= 2 
7 ran 6 pa 
+ an {Q„N n — M ;/ R// — M„R» — 
(N„Q„ - M„R»).(vii.). 
7 \nrcdff + aV 
5. The couple vanishes when for all values of n, 
M» 
Q» 
7T = tan <f> ni say, 
so that H 0 is of the form 
2 [Q„ cos n6 + R„ sin n0~\ cos (^ ^ 
in other words, the couple vanishes when both parts of each harmonic are in the same 
phase, though that phase be not the same for ail harmonics. As a particular case the 
remarkable result holds that whatever be the nature of the external field (it being 
made up, of course, of currents parallel to the axis), there will be no couple on the 
shell, provided the external field be altogether in the same phase. 
The Effect on an Infinite Cylindrical Shell, in the presence of an Alternating Current 
in a Parallel Wire, of the Interposition of a Parallel Cylindrical Shell. 
6. Take the plane through the axes of the shells as that of ZOX, and a perpen¬ 
dicular plane through the wire as that of YOZ. 
2Q2 
