TKOFESSOK G. II. DAKWIX OX ELLll’SOlDAL liAKMOXlC AXAEYSIS. 545 
Next we have from (83) 
~=i d+/3h cos-'- 6) (W. 
ll 0 J ^ 
rroceediuc,’ as before, 
: = ia„ 7 '(r+l) 
to 
X 
H 2 
/3 4r+o 2r+l ^ /3 \ 
1 - ^'+-2r-1 - J(T:rr)) 
LV ' 8r(r-l) 
+“ 0 / (1) [1 H-4^+Tti/5’]+“o^^K— ifo/3’). 
Substituting for h its value (80), rve tiiid that the term of order zero is (/'+!), 
0 
2 
and by (92) this is equal to 
The term of the first order is 
/3^«2./(^’+1) 
•> 
which may be written in the form 
1 , ^+f 
"• O 
U’ 
2/3 
| 8 Xa. 2 ,.y(r+l)-}-/3ao) and is equal to ~— 
0 2 (,+ I 
The term of the second order is 
4r + 5 
^-^« 2 ,.y(a+i) 
_8r(r-l) 4r(r-l) ' Si 
2r+l , 2r + l 
H—^o.r (^ + 3) 
+/5^(y-A)+'8^/3X, 
w 
hicli is equal t 
o 
0-+3) 2a„r(r)+/3-^«an--A)+ 
Hence tlie term may be wiitten 
Now 
.-I 
/•+ 1 - ... 
1) + /(^'+ (i:/ ~ i'-.>)+"h'^'«u 
21/0 
But f(r) = ( — )'■ 7 --— T~ ' , 
' 7 ' 1 )=-(-)'■ 7777777[*('+ 77;i 
(-)' 
,/ 
(-)'• 
t + rl 
(rVfr • (/•+iy ' (r!y(r + l)J/-/-! 
4 A 
VOL. CXCVll.-A. 
