220 



PIERCE. 



/ 



dZ \ sin2 U sin .A cos 2) }= J - I Jo (2 A sin ^P), (63) 



where Jo is the Bessel's Function of the zeroth order, with a develop- 

 iment of the form 



Jo (x) = 1 - I. + 



22 ' 2242 224^6^ 



+■ 



(64) 



Before substituting in (60) let us simpUfy the general trigonometric 

 factor in the brace of (60) by placing cosV by 1 — sin^, and letting 

 k = 2 A, as in (42), we then obtain 



- P r^" \ 1 - Jo ( A: sin .A) M- • , , • ^ „ 



^ = Ici \ sin^f i^-sm^^sm^^ 



] 



— 2 cos £ cos {B cos '/') — 2 cos yp sin B sin {B cos yp) r d\{/ 



^P f' \ k^ sin^ i/^ _ k^ sin^ 1/^ k^ sin^ i^ _ ^ 



22 



2242 



22 42 g2 



f 



j 2 — sin2 xp sin2 B — 2 cosB cos (5 cos i^) 

 — 2 cos ip sin B sin (fi cos \p) r d\p. 



or 



(65) 



P = 



4c 



-22:(-i) 



5- / sin"' 



• ■ -n- Jo 



-\- 2 cos BZi- 1)^ 



22425; 



^ ^ cos (i5 cos \p) d\p 



+ 2 sin 5I(- 1)" .,,,i".,^, JW-^ ^ cos ;/. sin 



224252 



w = 2, 4, 6, . 



(fi cos l/') (/)/' 



(66) 



