An introduction to the differential and integral calculus and differential equations - Tapa blanda

Sinopsis

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. 1899 Excerpt: ...(calling this angle + 6), we obtain the first equation. To find the pedal equation, we have, from the first equation, m log r = m log a + log cos mO. Ml)" Differentiating,--=--m tan mO, r 1.-= tan mO, r r pr r r r, p = r cos m&, and amp = r.am cos mO = rm+1, which is the pedal equation. The equation of the pedal can be written down by the rule of Art. 228, and is amr = im+2/p'»+ or ay1 = r2m+1, which is of the same form as before, except that m is changed to m'(m + 1). Hence the polar equation of the pedal is m n 233. This can also be obtained as follows:--"VVe obtained above--Since the pedal is the locus of T, we must find a relation between p and ff, which we can do by eliminating r from (1), (2) and the equation rm = am cos me. Thus pm = rm cos'" mO = am cos"'+1 mO = a'" cos"'+1--j--ff 1 m + 1 r m + 1' and changing p and ff into r and e, we get the same as above. t This caution is not necessary in the curves rm = a'" sin m$, as r increases with 9 in this case. two perpendicular tangents as axes, and the pedal is ax by x2 + y 238. The above method applies to the general equation /!«) = o (i) The result of eliminating x and y from (1), (2), and (3) is the;.quation to the pedal. 239. Radius of Curvature. Def.--If the normals to two infinitely near points, P and Q, n a curve meet in E, then E is called the centre of curvature, and EP the radius of curvature, its length being denoted by p. 240. To prove that p = dsjcty. In the figure of Art. 220, since PQ may be regarded as the arc of a circle, centre E, we have Z QEP = QPjEP; and since TQEP is concyclic, Z QEP = Z T = df.:. dj/ = ds'p, or p = (kitty. 241. Intrinsic Equation to a Curve.--This is the equation connecting the quantities s and s being the length of...

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