The elasticity and resistance of the materials of engineering . ed by introducingthe form of the hyperbolic sine, and this may be done byadding and subtracting the same quantity to that alreadyunder the sign of summation, in such a manner that: u = ^[A„ sin {nr sin cp) . siJi (jir cos cp) ?\-y2A^sin(nr sin cp) e-^^ .... (52) . Now if the product: sin {nr sin q)) e~^^^^ be developed in a series and multiplied by A„, one term willconsist of the quantity : — r^ sin cp cos cp multiplied by a constant, and if: ^A^ sin (nr sin cp) e-^ ^^ ^ be replaced by simply : — ar^ sin cp cos cp all the conditio
The elasticity and resistance of the materials of engineering . ed by introducingthe form of the hyperbolic sine, and this may be done byadding and subtracting the same quantity to that alreadyunder the sign of summation, in such a manner that: u = ^[A„ sin {nr sin cp) . siJi (jir cos cp) ?\-y2A^sin(nr sin cp) e-^^ .... (52) . Now if the product: sin {nr sin q)) e~^^^^ be developed in a series and multiplied by A„, one term willconsist of the quantity : — r^ sin cp cos cp multiplied by a constant, and if: ^A^ sin (nr sin cp) e-^ ^^ ^ be replaced by simply : — ar^ sin cp cos cp all the conditions of the problem will be found to be is equivalent to putting: — ar^ sin cp cos cp 62 TORSION IN EQUILIBRIUM. [Art. 10. for a general function of r sin cp and r cos cp. This change willgive the following form to u^ first used by Saint-Venant: u = ^A^ sin (nr sin cp) . sih {nr cos cp) — ar^ sin cp cos cp . (53) «Fig. 4 represents the cross section with C as the origin of co-ordinates, and axis. The angle q) is measured positively. from CN toward CH. At the points N, H, K and Z, in theequation to the perimeter, dr^ will be zero. Hence, at thosepoints, by Eq. (21) : -r = ^\An ^^fi (^^^ si^ (p) • ?^ cos cp . coh (nr cos cp) -\- An . n sin cp . cos {nr sin cp) . sih {nr cos cp)] — 2ar sin cp cos cp = o. At the points under consideration cp has the values 0°, 90°,180°, 270° and 360°. At the points A^and K, cp = o or 180° ; Cll£ hence, sin cp = o, and both terms of the second member of -— dr reduce to zero, whatever may be the value of n. But at H and Lj cp = 90° and 270° ; hence, sin cp = -\- i or — i and cos cp = o. In order then, that -,- dr Q ^.t H and Z, these must obtain: Art. lO.] TORSION IN EQUIIIBRIUM. 63 COS nr = cos (— 7tr) = o. If HL = c; and, KN = b; then : no f nc^. , , cos— = cos[-—)=o (54) If the signification of n be now somewhat changed so as torepresent all possible whole numbers between o and 00, Eq. (54)will be s
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