PiiOFESSOIf a. 11. L)A1{^\■IX OX ELLIPSOIDAL IIAPMOXIO ANALYSIS. 
52:3 
27rM i + -S’; 
'li 1 i ~ s\ 
+ A/3-[2S~ _ X + G - ^^(22 - I) + TJ} 
3 1 _ 4 . -A.^[2X^ + 3X - - I) + T] ; 
.... (71). 
'fhe following results may Ije obtained either by direct develoj)ment, 
or 
1)V 
inultiplicatioji l)v either or l)oth the factors 7,0 and Tlie former convert 3 
into P, the latter (>7 or S iuto C or S. 
^s) dcr 
iDP/lSffdo- 
27ri\I i + s I 
2lP\ r-~s\ 
'.{i-m 
+ 7o/5a2V + s - 6 - - I) + Tj;, 
;i + + -) 
+ -A'/SVII- + in + “■(■•IV - 22 ; + 1) - Tj;, 
(1 + 4/3(5 + 2) 
+ -,V/S-’[95 + 4 + .7(2S^; - 25 + 1) - T]i 
(71). 
15. Inteyration in the case of s = 2. 
Althougli the development in powers of sec- 9 is still legitimate in this case, yet 
the formuke fo\md in the last section fail because T contains 6-’ — 4 in the denomi¬ 
nator. Moreover since is not convertible into P- by a factor each case must be 
considered separately. 
\Ve noAv have = ainl therefore from (G8) 
= 7 ^ ~|l P 2/3 
% 
+ /3- 
-2(ifi - 2)u - ;3) (7 - 4) (7 - 5) + - 
+ {([ f (^' + v) (< + (i " vj + 27 y^^ 
il — 1) 7 ( / + 1) (/ -f 2) 
(/_2)(7-;;) 
(7 + 1) (7 + 2)__ 
+ i/3v*4-:n/3(s + 1 ) - cos 2 ,/,;. 
? — A I 
/AdP is e(pial to the same with accented 7 ’s, and the last term ecjual to 
{2 + d- P) — cos ’2(f) !j. 
'3x2 
