PROFESSOi; G. ir. DAPvWIX ox EI.LIPSOIDAL HAPvMOXIC AXALYSIS, 
Jn this formula .s is a constant integer and cr a constant to be determined. 
Our e(|uations are now 
[Df — i [f d- 1) [m(! — fi) — 1] — A- + /3aj 
a similar e(|\iation for g, 
and [D/ — f (i -|- 1) cos '2cf) s' — /3a] Ug = 0 
= 0, 
. (3). 
And Laplace's equation is 
' 1 — /3co,s2^\ , fl — /3Qus2(f) 
L'lLLUo = 0. 
1 •• o 
. . . . (4). 
Laplace's operator is equal to the differential operator in (4), divided by 
1 — /S cos 2(j) > I'l — 13 cos 2(j) 
- Irir/ - 
13 
0 
It is well known that in sj)heroidal harmonic analysis there are two kinds of 
functions of v and p which satisfy the differential equation, and they are usually 
denoted Pf, Qf. The Q-functions of the variable p have no significance, so that 
virtually there are P- and Q-functions of v, but onh^ P-fimctions of p. The like is 
true in the present case, however, with the additional complication that each of the 
functions may assume one of two alternative forms. T adopt a parallel notation and 
write for Uj and either 10.', or P.', Q.', as the case may be. Since v and p 
enter in the first two equations in exactly the same way, we need only consider one 
of them, and we may usually write simply (for example) ^3/ "^'diere the full notation 
w'ould be ^>1’ P-)- Ill llie early part of the investigation I shall only refer to the 
P-functions, and the Q-functions will be considered later. 
In s]dieroidal harmonic analysis the third function is a cosine or sine of .v®. So 
here also we find functions of two kinds associated with cosines and sines, which T 
shall denote (Lf, Sf, C/, Sf, the variahle <b l)eing understood. 
Throughout the greater part of this paper the functions will be of degree denoted 
by i, and it seems useless to print the subscript i hundreds of times. I shall accord¬ 
ingly drop the subscript i except where it shall be necessary or advisable to retain it ; 
for example, jT’ will be the alnidged notation for (e). 
The operators involved in the difterential equations (3) will occur so frequently 
that an abridged'notation seems justifiable. I therefore write 
— Ifi" “ I' (I' + l) [^' (1 — — 1 I — ff- / 3 tr , 
X,5 = Dg" — i {i + } ) /3 cos 2(f) .P — /3a , 
where a. = (I - (h’ - (e= - If i. 
a, = (1 — ^cns 
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