PKOFESSOli G. H. DARAVIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
477 
/ I cos , \ 1 „ I — t~) \ cos , . . . , , r cos . , , 
X. (L,j-+ 
sin 
+ ^ sin {sin^^ ~ 
(S.) ToJin<lxs[<^\^t4> )• 
sill 
1 now use the foi'iii as defined in (5), where l).j is given in (2), so that 
T, , ^ , 1 d d 
We have 
^ {sh/"^) “ ^ ^ {cos^'^ + -)^ ± — 1) — 2)(^ 
and 
cos 
- r-{V,; t4, + -i0(t+ t)(I+ 2 )];(/, + o 4> 
cos 
sin 
+ - I)(t 
CDS 
sm 
(/ - 2) cf) 
The latter terms of y^s contribute 
cos 
(R \ {s- — /3o-) <] t p — {i -\- 1) 
cos , , , . cos , , 
. ^ + 2 </,+ . {t- 2)(^ 
sin ^ ^ sin ^ ' ” 
Therefore 
/ r cos , \ 
X>!<1>L: '•#') = 
_ I 
/3i> i- ■*' ^ ' I™ '<;> + [>,t + {“h< + i)4' 
. ^ [ cos , , , . , 1 r cos , 
+ 2o- W tf/) + {o i' — I N . (/ — 2) 
sin ' ' '■ sill ^ 
(15). 
§ G. Deferminntion of ihe Cooffcicnfs io f/ic Fiiviclioiis. 
In this section I use successively the four results (12) (13) (14) (15) obtained In 
the last section under the headings (a), (/3), (y), (8). 
(oc) = 7.P -f d- 
The limits of the first 1 are 1 to d.*? or ^ (s ~ t), and of the second 1 to I {t — s) 
or i (i — -s — 1). 
Applying the operation ifj^ to ^1'’ and equating 
/3 
yjjs (P') to zero, we have 
