. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Curtate cycloid. Examples. 1. Show that the curtate cycloid cuts the axis of x at right angles,and, in general, that the line RP is perpendicular to the tangent tothe trochoid at P. 2. Determine the points of inflexion in the prolate cycloid. Showthat at the point of inflexion the radius of the generating circle is tan-gent to the curve. , b ib = cos-1— . d\ . a To find —4 in terms of tj\ use the method exemplified in Art. 289. § XXXL] THE EPICYCLOID. 313 The Epicycloid. 293. When a circle, tan-ge
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Curtate cycloid. Examples. 1. Show that the curtate cycloid cuts the axis of x at right angles,and, in general, that the line RP is perpendicular to the tangent tothe trochoid at P. 2. Determine the points of inflexion in the prolate cycloid. Showthat at the point of inflexion the radius of the generating circle is tan-gent to the curve. , b ib = cos-1— . d\ . a To find —4 in terms of tj\ use the method exemplified in Art. 289. § XXXL] THE EPICYCLOID. 313 The Epicycloid. 293. When a circle, tan-gent to a fixed circle exter-nally, rolls upon it, the pathdescribed by a point in thecircumference of the rollingcircle is called an epicycloid. Taking the origin at thecentre of the fixed circle,and the axis of x passingthrough A, (one of the posi-tions of P when in contactwith the fixed circle,) a, b, tp,and Xj being defined by thediagram, we have, evidently,. Fig. 60. *f = &X-:x = -£ f- The inclination of CP to the axis of x is equal to tp + X> or *0 - b ;— ip ; the coordinates of P are found by subtracting the pro- jections of CP on the axes from the corresponding projectionsof OC; hence / 7\ 1 7 a + b t x — (a + 0) cos ip — b cos —-— tp y — (a 4- b) sin tp — b sin b a 4- b r (1) These are the equations of an epicycloid referred to an axispassing through one of the cusps. 3H CERTAIN HIGHER PLANE CURVES. [Art. 293. Were the generating point taken at the opposite extremityof a diameter passing through P in the figure, the projection ofCP would be added to that of OC; the axis of x would in thiscase pass through one of the vertices of the curve, and thesecond terms in the above values of x and y would have thepositive sign. Algebraic Forms of the Equations. 294. When a and b are incommensurable, the number ofbranches similar to that drawn in Fig. 60 is unlimited, and thecurve is transcendental like the cycloid, since th
Size: 1569px × 1593px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1870, bookpublishernewyo, bookyear1879
