PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
153 
this becomes infinite for £ = co . The second solution is 
qn{V)=Pn{V) f 
dz 
h{p«(X)Y^ + i) 
= (-)”*! Vn (0 arc cot £ - 2 -~ ~ (£) + y } p«- 3 (0 
£-«-] _ + !)(»+ 2) £_„_ 3 
m. 
1.3..(2» + 1) 
+ 
2 (2% + 3) 
(» + !)(% + 2) (A + 3) (% + 4) 
2.4 (2 n + 3) (2 n 4- 5) 
£-«-S _ 
, * (?8) 
the latter expansion, however, being only convergent when £ > 1. This function 
q>i (0 vanishes at infinity. 
Hence we have the following solutions of (75) : 
Y = P n (fi).p„(Q 
V = P„ (n). q n (£) 
(79) 
the former being appropriate to the space inside the ellipsoid, the latter to the 
external space.* 
In like manner, when V involves co we have the solutions 
v = (1 - ( p + !)« rf U^Acosj sco 
dp? 
^(i-cr^rP + ir 
dX s Sill 
d s q„ (£) cosl 
dX s sm 
i 
(80) 
It seems unnecessary to go through the details of the investigations corresponding to 
those of §§ 10, 11. The results are, for the zonal harmonic, 
r = 
n (n + 1) p 
and, for the tessaral harmonic, 
9 = C . P„ (p), 
& 2 £ [Y %\ v?, d P“ (to) dt ln (So) . 
4°(C 0 +1) ^ ^ > 
<£ = C.(l-“ /x 2 )^ 2 - y ;^ - sill SCO, 
= (-) 
dp s 
s - 1 4?r l 7t ~ s M?n 2 _+ 1 f _ 
dtn(Q du n s (£ 0 ) 
p k + s n (n + 1) (£ 0 3 +1) — s 3 
o 
(81) 
(82) 
* The second solution is finite even when r = 0, but its space derivatives are infinite at the focal circle 
x> + y* =ld, z = 0 . 
MDCCCLXXXVII.—A. X 
