220 



PIERCE. 



/ 



dL \ sin2 (^ sin ;A cos 2) I = ^ - I Jo (2 A sin ^), (63) 



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

 ment of the form 



X- 



Jo (x) = 1 - - + 



.T" 



22 ' 224- 2^42 62 



+ ■ 



(64) 



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

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

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



P 



4c Jo ( 



— Jo (k sin \l/) \ 



sini/' 



2 - sin'xjysm^B 



— 2 cos B cos (B cos 1/') — 2 cos ip sin B sin (B cos \}/) \ d\l/ 



} 





i 



F sin^ i/' k^ sin^ \}/ k^ sin^ \p 



} 



22 



2242 



22 42 g2 



■| 2 — sin^ xj/ sin^ B — 2 cos B cos (5 cos 1^) 

 — 2 cos 1/' sin B sin (5 cos i^) r d\}/. 



\ 



(65) 



or 



V 



4c 



-22:(-i)^ 



22425; 



62---n2 Jo 



+ sin2fi2:(- 1)^ 



2242(32 



^ ^ / sin"+i .A# 



^- -n- Jo 



+ 2 cos 52: (- 1)^ ^242(3^"..^^, j['''sin"-^^ cos (5cosiA) # 



-+ 2 sin 52: (- 1)^ 2HW---7i^ Jo ^'''""' "^ ''''^ "^ ^'"^ 



(5 cos \f/) d\l/ 

 n= 2, 4, 6, (66) 



