Differential Calculus for Beginners
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ... Polar Curves. We shall next reduce the formula to a shape suited for application to curves given by their polar equations. 1 du 1 therefore p =-T---, 130. Tangential Polar Form. Let the tangent PT make an angle yfr with the initial line. Then the perpendicular makes an angle a = yfr---with the same line. Let 0Y = p. Let PaP2 be the normal, and P2 its point of intersection with the normal at the contiguous point Q. Let 0YX be the perpendicular from 0 upon the normal. Call this pt. Let P2P3 be drawn at right angles to PiP2, and let the degrees respectively, they will cut in mn points real or imaginary. length of 0F2, the perpendicular upon it from 0, be p2. The equation of PXT is clearly p = x cos a + y sin a (1). The contiguous tangent at Q has for its equation p + Bp = x cos (a + Sa) + y sin (a + Sa)...(2). Hence subtracting and proceeding to the limit it appears that dp-T-=--x sin a + y cos a (3) is a straight line passing through the point of intersection of (1) and (2); also being perpendicular to (1) it is the equation of the normal PiP2. Pp_ da?' Similarly represents a straight line through the point of intersection of two contiguous positions of the line PtP2 and perpendicular to PiP2) viz. the line P2P3, and so on for further differentiations. From this it is obvious that 0Y=dp dp ety. Uri da d+' da' nv--dL etc. Hence P1F=, ay and p = P1P, = OY+OYt=p +...(i). This formula is suitable for the case in which p is given in terms of yfr. Ex. It is known that the general p, equation of all epi-and hypocycloids can be written in the form p = A sinlty. Henee p = A sin Btj/-AB2 sin Bj/, and therefore p p. 131. Point of Inflexion. If at some point upon a curve the tangent, after its cross and recross, crosses the curve again at a third...
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