14 
Also 
Hence 
PROFESSOR G. H. DAPOVIN ON ELTJPSOIDAL HAR^FONIC ANALYSIS. 54:] 
1-1 I* 
Introducing the value of F as defined in (79), we have L or 
+,7m Ar:[i/3-i/3^], 
We have in (74) obtained ji^(^i^)"[(P;^)' —(Pd)'']do-, and if it he added to onr last 
result we see that the term which does not involve tlie factor l/(2/+l)is aimiliilated. 
and 
|p(i;'S,‘)Vo-=|^^:[l+i/3(,/-2)+jij^-(/-->C/+48)] . . (fU), 
Now from (;37) the square of the factor for converting into S‘ is 
IT. (®'“) 
1) 
^ = i+p+^;8^(./-s). 
Therefore 
T'hese are two of the rec[uired integrals. 
2'7rM / P1! 
2i + ].' 7 — 1 ! 
^ +iy^ {.) -f - ) +( 7 '+ I 9/ — 9hi) 
(94). 
Next we have from (S:]) 
51 
(cos-''co.Y-'' 6) ^(10. 
't 1 j 
2,+l 
Noting as before that f('r) = ~ /(7'd-l), nnd using the integrals (91), 
51 
(.] 
T = Sy2/_2./X'’+ ') 
+yo./'(2) 
■ B (4r-l)^^ 2r+l / ^ 
^“^2/“ SFr-lV”^ 2r [ ■^2(r-l)j 
* H“(' ~^^) 
Substituting for g its value from (80), 1 find the term of order zero to be 
/. / I . \ 2 7 + 1 ! 
72 ,- 2 /(’’+ 1 ), or — 
27 + 1 7-1! 
The term of the first order is Sy 2 ,_ 2 ./’(^’+1) 
1 2/'+r 
2r"^ 4r ” 
+^ 70 / (^) (f+i)* 
