PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
291 
where 
M= forv 7 (Aft) ^ cos 2 ) F + 2E}, 
sm y c ' 
2\/ (Aa) 
“ y = v /|(i + a?—}' 
and F and E are complete elliptic integrals to modulus sin y. 
Also 
dM _ 
db 
\/ (Aa) 
b sin y (2F — (1 + sec 2 y) E}, 
so that the repulsion is 
r [2E - (1 + cos 3 y) F] [2F - (1 + sec 3 y) E], 
If the coils consist of two circles of radii a and c (the former the greater) with their 
centres coincident and planes inclined 9, we have from Maxwell (§ 697), 
M =[ p '> (°>I p . («) + ••• + rrdrr; (b) PMo)] 3 P, (0) +... 
r (r + 1) \ct 
= 4 A {o! p »w + oO‘®‘ p *< 9) + --- 
+ 
r.r + 1 \a 
3.5., 
where r must now be odd. 
The couple tending to increase 6 is 
_2.4...(r + 1)_ 
PrW-.. 
or 
1J2 - 2 
2 S 2 + ^ 2 N 2 ° 
1 MNp 2 dU 
^ S 2 + p 2 N 2 ’ 
1 c n 1 (c\ /3\ 2 5 cos 3 # — 3 cos 9 
OS^-TOW 2 -2- +••• 
X 
or 
27t^ 2 Nc 2 e 2 . 
S 2 + ^ 2 N 2 a 2 
sin 6 cos 6 
1 + I - ) (10 cos 2 9 — 3) -f- 
1 e „ 1 /c\3 /3\ 2 15cos 2 ^ -3 
1.2 a + 3.4 W \2/ 2 
sin 
2. In the course of the following work it will often be necessary to know what kind 
of distribution of electric currents is likely to be set up in conductors of various shapes 
on the introduction of external fields : it is known that in a sphere no external field 
can give rise to currents that do not circulate in concentric spherical surfaces, and it 
2 P 2 
