. The strength of materials; a text-book for engineers and architects. -. (1) * See, for instance, C. Smiths Treatise on Algebra (Macmillan), p. 383. CURVED BEAMS 541 R + 9 , r h dy _ -p /2 r cos 0 d {r sin 0)1/ R+^~ y R+TsiiT^^ + ? = 2r2R / C0S2 ^ ^ ^ R + r sin 61 (2) _ cos2 e _ (l-sin^^) _ 1 __^( • 0_ I^sin^ R4-rsin6 R+r sin ^ ~ R + rsin(9 r\ R + r sin ^ 1 sin ^ R / sin 0 R + r sin1 r sin ^ R + r sin 0 r R2\ 1 + r VR + ^ sinR/, R R + r sin r^/ R + r sin y • cos^ OdOR + r sin ^ + 2 sin^ R2 R2\ _d^r^ / R» + r sin (3) + sin OdOr + + (4) + Now - f4^^ ^^ + f RdO ^2 cosO 2 + R^- = 0+^(4a)

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. The strength of materials; a text-book for engineers and architects. -. (1) * See, for instance, C. Smiths Treatise on Algebra (Macmillan), p. 383. CURVED BEAMS 541 R + 9 , r h dy _ -p /2 r cos 0 d {r sin 0)1/ R+^~ y R+TsiiT^^ + ? = 2r2R / C0S2 ^ ^ ^ R + r sin 61 (2) _ cos2 e _ (l-sin^^) _ 1 __^( • 0_ I^sin^ R4-rsin6 R+r sin ^ ~ R + rsin(9 r\ R + r sin ^ 1 sin ^ R / sin 0 R + r sin1 r sin ^ R + r sin 0 r R2\ 1 + r VR + ^ sinR/, R R + r sin r^/ R + r sin y • cos^ OdOR + r sin ^ + 2 sin^ R2 R2\ _d^r^ / R» + r sin (3) + sin OdOr + + (4) + Now - f4^^ ^^ + f RdO ^2 cosO 2 + R^- = 0+^(4a) A • f d^ r Again / ^ , . ^ = / ^ y R + r sm (9 / __ dO^ K + 2 r sm . cos ^^ do sec^ n (^ ^ R + 2 r tan ^ . cos^ ^ R sec^ ^ + 2 r tan ^- seC / 0 dO (put tan ^ = it, then R(l + tan2 j +2rtan2 du = Jsec^ s 542 THE STRENGTH OF ^lATERIALS du 2R 2R dudu J R (1 + «2) +%ru ^ J R + 2 r M + R m2J { d u u + ^) +(l R2 2^R ?-(.?-£)? (5) This is of the form /d X ?/ dO R ^ rsin^ R /: + 13 P tan tan X — a U -r R \ ^ R2 — B WTien ^ = 1. tan = 1 = w ^ = — ^ tan

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