470 
PROFESSOR G. H. DARWIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
since this only amounts to dropping constant factors which may he deemed to he 
included in the, as yet, undetermined coefficients of the several series. 
I will now consider in detail the forms of the several P-functions of v (those for ^ 
following hy symmetry), and at the same time indicate more precisely the nature of 
the notation adopted. 
In the following series, indicated hy S, the variable t is supposed to 'proceed from the 
lower to the upper limit hy 2 at a time. The reader will he able to perceive the 
manner of the formation of the functions when he hears in mind that 
Aj = - 
1 + /3d 
R _ 7. 
(w - l)i, C, = 
1 -/sr 
— /i ' 
Type (4 or EEC ; 
w = 
Sa/ {v~ - 
0 
-1)^ 
Type AB or EES ; 
p^ == 
[v~ - 
2 
Type A or OOC ; 
jDi’ _ 
(ffi - 
1 
_ l)D^-i)('^e _ 
Type B or OOS ; 
w = 
'icL, (k- - 
1 
- 
Type C or OEG ; 
w = 
{ir 
1 
Type ABC or OES 
; P^ = 
? 
'^afV {y'^ 
- - 
Type CA or EOC ; 
P^ = 
"^a/v (w 
o 
_ - 
Type CB or EOS ; 
W = 
i 
"Erx/v (k' 
- 
1 + I3f 
1 ■ 
1 + /3 G 
1 -13! ' 
1-/3 
1 
1-/3 
Observe that P is always associated with ( v~ — 
1 - 
and that, each form being 
repeated twice, there are two forms of function of each kind. Moreover, a cosine 
and a sine function are always associated with different kinds. It is obvious that the 
-functions are expressible in terms of the ordinary P-functions of spherical 
harmonic analysis, and that if we take out the factor | —^) the P-functions are 
similarly expressilfle. This factor will occur so frequently that I write 
n{d) = 
/ o 1 + ^ 
\v^ -I 
and as elsewhere commonly put P to denote P(r). 
