Elements of analysis as applied to the mechanics of engineering and machinery . ill increase by i? ^ = 8?/, and s, by the element P Q::= 8s; and we have, in accordance with the Pythagorean theorem, PQ = P~P + Q R\ i. e.: 8s^ = dx + 8?/;therefore, 86 = ydx- -\ dy\ and consequentlj^, the arc itself; s Vdx 4- dy\ Art. 32.] ELEMENTS OF ANALYSIS. 49 For NeiVs parabola^ for example, (yicI. Art. 9, Fig. 1*1,) the equa-tion of which is ay- = x^, we have 2aydy = Sxdx; hence: according to which, Sx^dx , ^ , 9 x dxand. cy^ = 2 ay 4: a^ y^ 4:a (^ + ia) ^^ ^ -/^+^>-?/o+^D»(^:) ^ -1 In order to find the
Elements of analysis as applied to the mechanics of engineering and machinery . ill increase by i? ^ = 8?/, and s, by the element P Q::= 8s; and we have, in accordance with the Pythagorean theorem, PQ = P~P + Q R\ i. e.: 8s^ = dx + 8?/;therefore, 86 = ydx- -\ dy\ and consequentlj^, the arc itself; s Vdx 4- dy\ Art. 32.] ELEMENTS OF ANALYSIS. 49 For NeiVs parabola^ for example, (yicI. Art. 9, Fig. 1*1,) the equa-tion of which is ay- = x^, we have 2aydy = Sxdx; hence: according to which, Sx^dx , ^ , 9 x dxand. cy^ = 2 ay 4: a^ y^ 4:a (^ + ia) ^^ ^ -/^+^>-?/o+^D»(^:) ^ -1 In order to find the constant which is here necessary, we willallow s to begin simultaneously with x and y. AYe thus obtain0 = 2T a y 1 ^ -f con., therefore con. =^ — /y ^j ^^^l for example, for the portion AP^, having the abscissa x ^ a:^ = 2\ci [i/(W — 1] = 1,^36 a. Fi. 43. If, further, the tangential angleQPR = P TM= a be introduced,there results also, Q R ^ P Q . PR = PQ COS. QPR,i. e.: dy = ds sin. a and dx = 8s cos. a,and therefore, not only tang. a = ^ (Art. 6), but also. M Ndy dx sin. a = —- and cos. a = _-; and, further, OS cs dy ^ = j t/ 1 + tang. -^-d x =j^^^^ =j ^^^. If, now, the equation between two of the magnitudes x, y, s, and abe given, we can then also find equations between two others of these magnitudes. If we have, for instance, cos. a =-. ,, Vc-\-s there is also; ex = OS COS. a = du and J l/c -4- s -^ Vc 4- s -^ Vu ^ Vc + 50 ELEMENTS OF ANALYSIS. [Art. 33. = ^/c^ -f s^ -f- con.; and if x and s are, at the same time, zero: X == Vc^ H- s — c. Art. 33. A straight line at right angles to the tangent P T,Eig. 44, is also normal to the jDoint of contact F of the curve -, be- Fig. 44.
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