PEOFESSOR G. H. DARAYIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
4 (PI 
If Ui, u^, denote the three roots of the cubic 
.» .» 
^ = 1 
o . o 1 7 -1 , ^ I o , „ o -^5 
ft" “f" 0" —|- 7/' ('■' ”f" 
it may he proved that 
X- = 
(«~ + Uy~) (fd + u.f) (rd + Uf) 
(/p _ ^^2) (c3 _ rd) 
and if-, may ])e written down liy cyclical changes. 
If for hrevit}^ we vulte 
A„~ = + a~, B,y = n,^ + Jr, Ch~ = + m, {n = h 2, 3), 
Laplace’s equation 1)ecomes 
d d \A 
ini - v^)(xxr „; fy . + (-s"- - <) ( VBX!, 
V, 
cl \. 
+ {if -yr)( A,B,C,^jY, 
The solution is V, = 
where U^, L^o, L^g are functions of n^, u. 2 , Ug respectively, and satisfy 
= 0 . 
d \2 
Ui — + '<']U 
n 
and two other ecjuations with suffixes 2 and 3, involving the same k, a constant, and 
the same i, a positive integer. 
If a, J), c are in ascending order of magnitude we may suppose vf to lie between 
— rd and oo , ur between — and — JA, and between — l)^ and — cd. 
If . 5 ^, cS,, 63 denote the three orthogonal arcs formed l)y the intersections of tlie 
three orthogonal quadrics, 
/_dq_Y _ (?cd - -?q-) {if - uf 
\ufvj ~ AdBi'CL 
and two other equations found by cyclical changes of suffixes. 
§ 2 . Notation ; 
1 now change the notation. 
limits of /3 so as to represent all Ellipjsoids. 
and let the three roots be defined thus :— 
o 700 
ILp — Yp"} 
r, , ^ 1 — /3 COS 2<i 
’A = 1 ■ ■. 
where v ranges from 00 to 0, p Ijetween +!,<■/> between 0 and 27r. 
