REFLECTIONS FROM CIRCULAR^BENDS 341 



Rpm = (2 /a) I (pip ^ — \)s'm(Trpx/a) sin (irinx/a) dx 

 Jo 



(Al-2) 



(Al-4) 



Tpm = (ep/a) I (pip " — l)cos{Trpx/a) cos {irmx/a) dx 



in which eo = 1, €p = 2, p = 1,2, • • • 



In (Al-2), (1.2-11), (1.2-16), (1.4-3), (1.4-6) we make the substitutions 



p — m = r, u = pi Tr/a — -wjl = p27r/a 

 p -^ m = s, V = pi T/a + 7r/2 = PsTr/a _ (Al-3) 



y = TTx/a, w = piT/a, p = .r + pi — a/2 = aiy + u)/7r 

 Introduction of the integrals 



Is = (I/tt) / [w'^(y + w)~" — 1] cos sy dy 

 Jq 



= (1/a) I (pip~^ — 1) cos (irsx/a) dx 

 Jq 



Ja = TT I — , — dy = IT sin iirsx/a) dx/p, 

 Jo y -f- u Jo 



K, = ^J r ^ dy 

 a Jo y -\~ u 



enables us to write 



Rpv, = Ir — la , Spm = —mOT^Js + Jr) 



Tpm = ep{Ir + Is)/2, Upm = mepar\J, - /,)/2 (Al-5) 



V^ = Kr- Ks, Wpm= ep{Kr + Ka)/2 



where Is and K, are even functions of 5 and /« is an odd function of s. 

 Co = 1 and €p = 2, p = 1, 2, 3, ■ - • . Since w and u depend only upon the 

 ratio pi/fl, the values of Is , Ks and Js depend only upon pi/a and the 

 integer s. These quantities are tabulated at the end of this appendix. 

 Setting y + u equal to / gives 



Ja = IT I sin s(t — u) dt/t 



J u 



= Tr[Siisv) — Si{su)] cos su — Tr[Ci(sv) — Ci{su)] sin su 



where Si and Ci denote the integral sine and cosine functions. Integrating 

 by parts enables us to express Is in terms of J, . Thus 



/ (y + w)~2 cos sy dy = u~^ — v~^ cos sw — x~^ sJ, (Al-7) 



Jq 



