146 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
showing that Z must be a zonal harmonic of order n, of the first or second kind. We 
thus obtain the solutions 
v = p„m. 
v = p„m . Q,(0 /’•••• 
where Q )t denotes the zonal harmonic of the second kind, viz. 
QUO 
P ”^j { {P.tfUMf 8 -!)’ 
* p -«) l0 « ~ TUT p ->«) - iA^T) p -«>-••• 
_ I" 1 y-*-i _j_ ( n + 1 ) + 2 ) r-»-3 
1.3..(2% + 1) 1 2 (2%+ 3) 
(% + 1) (n -f 2) (n + 3) (n + 4) 
2.4(2% + 3) (2% + 5) 
The former of the solutions (47) is appropriate to the space inside the shell, the latter 
to the external space. 
In like manner, when V involves <y, we have the solutions 
V = (1 - 11 *) 
2W2 n (A*-) / jr2 _ _ 1 \sl 2 (^) COS'] 
(Ifjf 
v = (1 - ur r}“ 
(£ 2 - 1)' 
sin 
fib 
sin 
i" I 
SO) 
. . . . (49 )t 
J 
10. Proceeding now to the problem of free currents, we shall show that the condi¬ 
tions for a normal type are satisfied whenever <£, considered as a function of p, and a>, 
* The following are the values of the first four solid harmonics of this type, expressed in terms 
of x, y, z :— 
&p x O) Pi (t) = *, 
FP 2 (/i) P 2 (?) = \ { 6 z 2 - 3 (»» + 1 /) - 2 k 2 }, 
* 3 P 3 00 P 3 (?) = fz {10z 3 - 15 (»» + i/O - 6 * 8 }, 
/OP 4 00 P 4 (?) = ^{280** - 840z 2 (a* + y 2 ) + 105 ( x 2 + y 2 ) 2 - 240£ 3 z 2 + 120 £ 2 (x 2 + y 2 ) + 24**}. 
| I have here only recapitulated, for purposes of reference, the principal steps in the deduction of 
the solutions (47), (49). For details see Heine, ‘ Kugelfunctionen,’ vol. 2, part ii., chap. ii.; or 
Ferrers, ‘Spherical Harmonics,’ chap. vi. 
The following are the values of a few of the more important solid harmonics of the form given by 
the first of equations (49) :— 
n = 1 , 
s = 1 
y- 
n — 2, 
s = 1 
9 xz, 
9yz. 
n = 2 , 
5 = 2 
9 ( x 2 _ 2 / 2 ), 
I8xy. 
= 3, 
5=1 
-fx {20z 2 - 5(x 3 + i/ 2 ) — 4fc 2 }, 
fi/ {20z 2 — 5 (a 2 + 1 / 2 ) — 4A- 3 }. 
n — 3, 
5=2 
2‘25z (x 2 - y 2 ), 
450.X2/Z. 
n = 3, 
5=3 
225 (x s - 3xy 2 ), 
225 (3.t- 2 2/ — ?/ 3 ) 
n = 4, 
5= 1 
-\ s -zx {28z 3 - 21 (x 2 + y 2 ) - 12 t 3 }, 
-\*-zy{28z 2 — 21 (x' 2 + y 2 ) — 12 A 3 } 
