706 Expansion of Bessel Functions of High Order. 

 with z, and if 



T , , /2R . . ' 



T , w , /2K A >• • • (35) 



■n— § V 7T2- 



m = 2?z + 1 = 2z sin a. 



J 



Then ^ , X s , X. - /il/t -; 



K=sec« + — — 2 .sec a+ 7 ^ N - 4 .sec°a+ ..., (36) 



where 



(5 + 3)X s+4 +(s + 2) 3 X s+2 + 2m 2 5.5 + 1.5-f 2.X s + m*,s.5 2 -4.X s _2 = 



X 3 =-i X g = J(27-24m 2 ), X 7 = ^ (1160m 2 -1125- 40??i 4 ) 



. . . (37) 



and every third term, commenceing with the second, is two 

 orders smaller than the one preceding. Moreover, 



. 7T / 7T \ OL f M2 Ms > \ 



^=j + ^cos«- f -« S m^-^-- 3 +- i ...J 



— T~ \ —v tan « — ^| (tan a — - tan 3 a ) + ^ 5 J tan « — 



|tan 3 « + i tan 5 a W . . . j , . . (38) 

 where the coefficients //. and X are connected by 



..( i +?+9+"-H i+ '5 + ^ + -T'- <*» 



and every third term of the brackets in <£, commencing with 

 the second, is two orders below the one preceding. 



In calculating results to a definite order, the coefficients 

 may be simplified. For example, if R is required to order 

 z~ 2 inclusive, it is sufficient to take 



^3 = — 2 > K — — % m2 > X 7 = — § m 4 , A 9 = 0. 



For most physical problems connected with the intensity 

 of shadow behind large spheres, the approximation to order 

 z~ 2 is sufficient. 



If 



^ = -r + z (cos a— -x — a sin a j, . . . (40) 

 it is found that to this order, on reduction after substitution 



