CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 185 



THIRD EXAMPLE. 



23. Consider the integral 



2 r " r" r* * 2 xi x<i 



v = — / / » cos (a? cos 6) x dm d0 = ' uxdx = ; — + — — — - ... * (57) 



it J a Jq "o 2 2". 4 2 . 4~. o 



which occurs in investigating the diffraction of an object-glass with a circular aperture. 



d d 



— x~' — 

 da das 



d d 



By performing on the series the operation denoted by x — — x~ x — , we get the original 



series with the sign changed, whence 



d*v 1 dv 



da? x dw ^ 



We may obtain the integral of this equation in a form similar to (49). As the process is 

 exactly the same as before, it will be sufficient to write down the result, which is 



v = J'ai {R cos x + S sin x) + B' coh (R sin x - S cos x), . (59) 



rhere 





-1.3.1.5 -1.3.1.5.3.7.5.9 

 R= \ — + 



S 



1.2(8xf 1 .2.3.4 (8.Z?) 4 



1.3 -1.3.1.5.3.7 



(60) 



1.807 1.2.3(8,f) 3 



the last two factors in the numerator of any term being formed by adding 2 to the last two 

 factors respectively in the numerator of the term of the preceding order. 



The arbitrary constants may be easily determined by means of the equation 



dv 



— = ux. 



dx 



(61) 



Writing down the leading terms only in this equation, we have 



xi {-A' sin x + B' cos x) = 71-"^ (cos x + sin x), 

 whence 



-A'=B' = ir-K 



v =(^f{ Rcos {*-T) +Ss ™{*-T)}' ■ ■ (62) 



24. Putting in the formulas of Art. 13, 



a'=-1.3; 6'= 1.5; c'=3.7; d'=5.9; e'=7M; D = 8 ; 

 we get 



4,= -3.8; A t = - 3\8 2 .11; d = - 3 ; C 3 = - 2 . 3\ 5 2 ; C 5 = - 3 3 . 4*. 5 2 . 127 ; 



£', = -3; -E 3 =-S 2 .8; E 5 = -3 3 .4.8 2 . 131 ; 



* The series 1 - -— - + ... or -= has been tabulated 



2.4 2 . 4 2 . 6 x* 



in his work on diffraction. The argument in the latter table is 



180° 

 the angle *, and the table extends from 0° to 1I2.V at in- 



by Mr. Airy from jr = to .r = 12 at intervals of 0.2. See Camb. I 



Phil. Trans. Vol. v. p. 291. The same function has also been , tervals of 15°, that is, from,r = to x = 19.63 at intervals of 0.262 



calculated in a different manner and tabulated by M. Schwerd ' nearly. 



Vol. IX. Paiit I. 24 



