474 
PROFESSOR G. IT. DARWIN ON ELLIPSOIDAL HARMONIC ANALASIS. 
,,T. , . ^/P* , I'~ I 
We have now to eliiniiiute v 7 - and —r . 
Jv ir — i 
It IS known that 
and 
The differential equation satisfied by P'' involves t in the form t~. Hence 
{ir — 1 )“-^—^ P can only differ from by a constant factor. In order to find 
that factor suppose v to be infinitely large ; 
tlmn 
and 
Also 
T. 2i\ . 
P zzr- 1 / 
P' = 
! 
l ! 
2A 1 i — t':' 
.3 _ n-p/AL\ p _ -ff- 
^ 2‘iilf i + tl 
^i + i _ 
2i ! P 
2b ! i + t\ 
% "f" t ' 
Therefore the factor is .-ff, and 
% — i 
U / d \ 
F = p-l)q-)P = ^-^P-l) 
x-ui±\-' 
dv 
P. 
It vnll be convenient to pause liere and obtain the corresponding formulae for the 
Q-functious. Various writers have adopted various conventions as to the factors 
involved in these functions. I write 
and 
As in the case of P' we may change the sign of t, if we introduce a constant 
factor, and this may be fouud by making v infinitelv great. In that case it is easy to 
show that 
2AVP 
2t + 1 ! 
1 
V 
i + 1 
Pv performing ( —) and ( —on O it follows that the constant factor is the same 
^ ' dv : dv; 
as before, and that the alternative forms for Q'' are exactly the same as for Pt 
Hence the transformations which follow for the I^-functions are ecjually applicable 
t'") the (:^‘-functions. 
