538 
PR0FE880E G. II. DAEWIX ON ELLIPSOIDAL IIAP5IONIC ANALYSIS. 
From this 1 find E or 
COS'" 0 . AflO: 
22"+i(7i!) 2 n-1 : 
9.; 
«+1: 
2^/1+ 1 
X(-V 
1.3P..(2/--1)^ (2r-fl) 
2>(h_ 1) („_2)...(»-r).r+l : 
_,y>+ 
K- 
(-y‘2n: (2/? +1) (2« + 3)^... (2/1 + 2.S--3F (2// + 2.s-1) 
2-«-^ // ' // + 1 : ^ ' 2-*-' (// + 2) (n + 3) ... (« + .A. .s - 1 : 
X 
1 1 H 
2 lou- ^ ■ 
^ K ^ t 
1 1 
.<-1 
2/—1 2 // a 2 -<; —1 
This is appiicahle also to the case of n = 0. provided that X interpreted as zero. 
0 
In the particnlar case in liand 1 find, hoAvever, that it is shorter not to n.se this 
_o*eneral formula, but to carry out the transformation (85) in the particular cases 
where the result is needed. 
§ 20. Beduction of jjveceding integrals; disappearance of logarithmic terms. 
In the application of the integrals of the last section, we are to put = 1 
and only to develop as far as /3~. 
'fhen to the proposed order k~ = 2/3(1 — ^), = 4^85 
vVlso 
2 log (1+/^) —log 
1 + /3 
It will now facilitate future developments to adopt an al/ridged notation. 1 write 
then 
, , . 2'’' n ! /t — 1 ! 
./ (n) = 
29 ! 
and observe that_/’(// -f I ) = /(l) = ./C^) = 3 - 
Since k' is of the first order in y8, only the first series in tlie D integral (84) enters 
when n is greater than 2. In that ca.se 
=/■(-) 
2(//-l)~'~8 (;//-l) pi-2)_ 
._ § (4'/i, + 1) _ 
2 (// — 1 ) ' 8 (// - 1 ) (?/ — 2 ) 
. . . (S6). 
4'liis result may he obtained very shortly Avithout reference to the general formula ; 
for when n is greater than 2 
1 ): 
p 
\l-sl l_, 
cos~" 6 dO 
\l-^l ]_,^(cos^d + 2/3 + 2/3-f 
= i 
1-/3/ )_u 
COS'" ^ 9 
I — 
3/3- 
d±f: _ 
’OS-2 cos'^ 9 
dO. 
