Elements of analysis as applied to the mechanics of engineering and machinery . ^^ T A M if 0 K Generally, « can only be determined by means of the equation of co-ordinates, since we put tang, a = -^, ox But there is, further, 8 tang, a = COS. a? dxand COS. a ?=--z^\ 8s ? 8 x^hence, we have 8« = cos. a? . 8 tang, a == —— . 8 tang. «, and G o 8s 8s^ ^ dx^ ds ds^ For a convex curve, we have r = -\- ^z-- = -{- -— 0(j^ cx^ c tang, a for a point of inflection, r ; and oo. For the co-ordinates AO = u and 0 G = v of the centre G ofgyration there is u = AM-\- HG = X ^ GP sin. GPH^i.
Elements of analysis as applied to the mechanics of engineering and machinery . ^^ T A M if 0 K Generally, « can only be determined by means of the equation of co-ordinates, since we put tang, a = -^, ox But there is, further, 8 tang, a = COS. a? dxand COS. a ?=--z^\ 8s ? 8 x^hence, we have 8« = cos. a? . 8 tang, a == —— . 8 tang. «, and G o 8s 8s^ ^ dx^ ds ds^ For a convex curve, we have r = -\- ^z-- = -{- -— 0(j^ cx^ c tang, a for a point of inflection, r ; and oo. For the co-ordinates AO = u and 0 G = v of the centre G ofgyration there is u = AM-\- HG = X ^ GP sin. GPH^i. e. u = x -{- r sin. «, andv= 0G = 31P — HP = y—GP cos. GPH, v = y— The continuous succession ofthe centres of gyration gives acurve which is called the evoluteof A P, and whose course is de-termined by the co-ordinates uand V. If the ellipse ADA^D^ , be brought into connectionwith a circle AB A^B^^ its co-or-dinates GM= X and MQ = ymay be expressed by the angleP G B = ^ at the centre of thecircle. We have, for example,. 52 ELEMENTS OF ANALYSIS. [Art. 33. X = CP sin, OF31= GP sin. BCP = a sin. <p and V = MQ = - MP =- GP COS. CP3I=b cos. cp.^ a a From this tliere results ^x = a cos, (fb(p and dy -= — h sin. (pd(p^ consequently, for the tangential angle Q TX = a of the ellipse: dy h sin. cp h tana, a = —- = — = tang. <f. ex a COS. (f a and, for its adjacent angle Q TG =^ o.^ r= \^{)^ — a: b ^ a tang. a^=z - tang. 0 and cotg. 0.^ = — cotg. (p. Accordingly, the suhtangent of the ellipse isMT= MQcotg. MTQ = y cotg. a^ = --- cotg. <p = y^ co^- 9, if y^ designate the ordinate IIP of the circle. Since, in the latter,the tangent P T stands at right angles to the radius CP, there isalso P TM = P G B = <f^ and hence, the sub-tangent of the same,likewise: MT= MP cotg. 31TP = y^ cotg. cp. Therefore, the twopoints P and Q of the circle and of the ellipse which have the sameabscissas, have one and the same subtangent have, further
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