. A treatise on the mathematical theory of elasticity . first place the relation between the twist of the rodand the measure of tortuosity of its strained central-line. • Let I, m, n denote 252, 253] OF A THIN EOD 389 the direction-cosines of the binormal of this curve at Pj referred to the prin-cipal torsion-flexure axes at Pj, and let V, m, n denote the direction-cosinesof the binormal at P/ referred to the principal torsion-flexure axes at P/.Then the limits such as Km {l — l)/Ssi are denoted by dl/dsi,.... Again let 1+ Bl,... denote the direction cosines of the binormal at Pi referred to t
. A treatise on the mathematical theory of elasticity . first place the relation between the twist of the rodand the measure of tortuosity of its strained central-line. • Let I, m, n denote 252, 253] OF A THIN EOD 389 the direction-cosines of the binormal of this curve at Pj referred to the prin-cipal torsion-flexure axes at Pj, and let V, m, n denote the direction-cosinesof the binormal at P/ referred to the principal torsion-flexure axes at P/.Then the limits such as Km {l — l)/Ssi are denoted by dl/dsi,.... Again let 1+ Bl,... denote the direction cosines of the binormal at Pi referred to theprincipal torsion-flexure axes at Pj. We have the formulae* lim Sl/Ssi = dl/dsi — mr + uk,lim Sm/Ssi = dm/dsi — nK+ It, «s,=0 lim Sn/Ssi = dn/dsi — Ik + vik. 8si=0 The measure of tortuosity 1/S of the strained central-line is given by theformula 1/t = lim [{Biy + (8my + (8n)=]/(Ss,)S 6*1=0 and the sign of 2 is determined by choosing the senses in which the principalnormal, binormal and tangent of the curve are drawn. We suppose the prin-. Fig. 45. cipal normal (marked n in Fig. 45) to be drawn towards the centre ofcurvature, and the tangent to be drawn in the sense in which Sj increases,and we choose the sense in which the binormal (marked b in the figure) isdrawn in such a way that the principal normal, the binormal and the tangent,taken in this order, are parallel to the axes of a right-handed system. Nowwe may put 1 = fcp = — cos /, m = kp — siny, h = 0, where p is the radius of curvature ; and then ^tt -/ is the angle between the * Cf. E. J. Eouth, Dynamics of a system of rigid bodies (London 1884), Part 11, Chapter I. 390 KINEMATICAL FORMULA [CH. XVIII principal plane (x, z) of the rod and the principal normal of the strainedcentral-line. On substituting in the expression for 1/2 ^ and making use ofthe above convention, we find the equation •(2) _dl 1^~ds/S in which tan/= —(«/«) (3) The necessity of introducing such an angle as/into the theory was no
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