Expansion of Bessel Functions of High Order. 703 

 Accordingly, 



t_Jl _1 _L ]0_ _i 56 280 

 ~~ \w W X 5 ™ 6 + X ; W \ 7 w H + \V X 7 w 10 " * ' ? 



which may obviously be expressed in the form 



i.-{. + ; 



B 3 £ _d^_ lOf 2 _d 6 _ 



3I3«Bwi a + 5iasam 5 6! B- 2 dw 4 



^ d 7 56c 8 d 8 280s 8 B 9 IJL 



+ 7ia«Bi» 8+ 8! d^dm 6+ 9! B^dm 6 'J*^ 



=D.i, • • • < 22 > 



Xw; 



where D denotes the operation in brackets. 

 Accordingly by (19), 



R= (I 1 + I,)<ty 



= - f " D I 9 JTi + 9 ^T— W 



ir J lz: cosn y — m 2.: cosh Y + mj 



_fc D ff 1 , 1 -) . v. 



7r ' J C2ccoshi/r— ?n 2,0 cosh i/r + m j ''' 



since the integral obviously satisfies all necessary conditions 

 for differentiation. 



IT 



The value of the integral is 7 - rTl ^rr- as in (15), and thus 



B= iz { 1 + |j D^ 2 + |j D.D/ + ^ D^D/ + i IW 



+ 5 fD 1 W + 2 ^ 3 lVD 2 e + ...} 71 i=, (23) 



where Di =^— , D 2 — ^ — • 



/yyy yi , I i 



If q-— — — '= sin a, the second approximation becomes 



juZ Z 



R = sec a. — £-5 (cos 2 a + 5 sin 2 a) sec 7 a. . (24) 



