PKOFJiSSOi; (P 11. DAKWIX UN EL1>TPS0I1)AL flAlUMONrC ANALYSIS. 
o2.-) 
Tlieu as far as material 
{€rf 
y 'I ■) 
(1 — /3cos '2'p)' 
(I —fS cos'2 cf)Y 
ill + /3' [(/a)’" + + po + i/A + T-2 
+ yS(2?j + ’-^Pt) + s) ) 
i (I + /3' [(/a)' + i/A + tV \+ /^{Pi 4- i) cos 2c/). 
N 
ow 
ku 8' 1A 2 J , 1 
= HU- H/saov 
(1 — /3 cos 2(^)" 
= - 
(1 - /Scos2(/)y- 
.0 
We ]unv multiply these by \Fidd and \F^d$ respectively, and the last terms dis¬ 
appear as before. I remark that the disappearance ot the terms which do not involve 
the factor 1 (2i + 1) atfords an excellent test of the cori-ectne.ss of the laborious 
reductions throughout all this part of tlie woi’k. 
Then we have 
I 27rM i-j-2!. 
_ 27riM < -f 2! ^ 
“21+1 /-2k 
- T/3(N -1) -h o b e ^ ■( 1 y _ 13 ON + 8 0)] 
X|:i+-A-/3~(19N--8N + 22 )] 
i/8N+.,f,.y8-('J5N--98N + 72)} . . . . 
(fb). 
If we multiply this by 
1) 
/(cos) 
01 
• 1 — '4(5/8” (5N + /), we ol)tain the result wjien 
C,‘ replaces C/" ; the only change is that the last term inside { f now liecomes 
+ .i(5i8”(!.)5N--ir8N-40). 
Again 
/_■)'(' ~^l"'[l8":i/^l- + b) + 7k^y^g25N-d-t8GNd-oG8)] 
x[I + ,4/3-(N-~8Nd-i8)J 
Urfl ,ii./3-(2»SH so^ + 21«)]. 
2/ + 1 /-2h 
If we multijily this by 
L.J>/ 
.(sin) 
. . . (72). 
or I -h i', 5/8” (- + 5), we obtain the result 
when S," replaces ; the only change is that the last term inside J ( now liecoines 
+ +k/3”{/klN’+ i06N+ IGG). 
T'his terminates the integrals, which can lie completely determined liy this method 
of developing in powers of sec’ 6. 
