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Mathematics. — • TIw, valaes of some definite IntegraL- connected 

 with Bessel f auctions'. By Dr. W. Kapteyn. 



The integrals referred to are 



271 



ƒ COS (x sin 6) — cos (x sin w) 

 ^ ^f^ -5^ ^ d 6, 



cos 6 -\- cos <p 







2- 



ƒ sin (.?; sin 6) sin 6 — sin {x sin <p) dn (f 

 -— d(9, 

 CSS a -j- cos if 



u 



2- 



ƒ COS {x cos 0) CSS (x COS w) 

 — ^^ ~ ^ —da, 



COS 6 -j- COS (p 

 



27: 



sin {x cos 6) cos 6 — sin (x cos (f) cos (f 

 'S = I ^— d 6 . 



cos -f- COS (fi 



-i- 



u 



If in these integrals we insert the wellknown equations 



COS {x sin &) = I, ^ 2 I, cos 2 ^ + 2 /, cos 4 <9 + . . . 



sin {x sin 6) sin 6 — I,{1 — cos 2 6) -^ I., {cos 2 ^ — cos 4 ^) -j- . . . 



cos {x cos 6) — 1^ — 2 1^ cos 2 6 -{-2 1^ cos 4. d — . . . 



sin {x cos 6) cos 6 — I^{\ + cos 2 6) — I^ {cos 2 6 -\- cos 4 ^) + . . . 



where Ip stands for the Bessel function J^ {,r) of order p, and if 

 we write 



27T 



ƒ COS 2 n 6 — cos 2 nip 

 ^-— dip 

 cos o ■-}- COS (p 

 



it is easy to find 



P=:7„J., + 2 /,.!, + 2 ƒ,.!,+ . . . 

 Q = A .1,, + (/3 - /,) .1, + {1, - I,) A, + . . . 

 i^ = -2/,.l, -h2/,.J,-2AJ, + . . . 

 S = I, A, - A (^1. + A^) + /,, {A, f J J ~ . . . 

 In order to determine A-2n I notice that 



COS 2n6 — co.s- 2nxp 



{a) sin <p — — =z — sin 2n(p -\- 2 sin {2n — 1) <f ros 8 — 



cos Ö -f- (-'OS ip 



— 2 sin {In — 2) ip cos 26 \- . . . 

 -|- 2 sin tp cos {2n — 1) 6 

 This formula can be proved as follows : 



