Elements of the differential & integral calculus

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. 1855 Excerpt: ...required for x and y in the value just deduced, substitute the co-ordinates of the particular point. As only the absolute length of the radius of curvature is required in determining the curvature of curves, we may use either the plus or minus sign of the formula. It is best, in general, to use that which, taken with the sign resulting from the expression, will make R essentially positive. Let it now be required to find the general expression for the radius of curvature of Conic Sections. Their equation is, i ' t. J (m + 2nx)dx.tr = mx + nr; whence ay = . '.--, 2y Inydx1--(m + 2nx)dxdy _ iny--(m + 2fw)te dy 2p-4? These values substituted in the formula, give R _ 4(mz + nx) + (m + 2bx)," 2n? and this, after dividing both terms of the fraction by 8, may be put under the form R _ (Vmx + nx1 + j (m + 2ru-)3)3. - T the numerator of which is the cube of the normal, Art. (82): Hence the radius of curvature at any point of a conic section, is the cube of the normal divided by the square of half Ihe parameter, and the radii at different points are to each other as the cubes of the corresponding normals. Tf in (1) we make x = 0, we have, at the principal vertex, R =--= one half the parameter, B' which for the ellipse and hyperbola is--. The radius of curvature at the vertex of the conjugate axis of the ellipse is obtained by substituting in (1), As R =---= one half the parameter of the conjugate axis. B It may be readily shown that _ is the least value which R admits of; therefore the curvature at the principal vertex of a co A nic section is greater than at any other point. Likewise, ---is the greatest value of R in the ellipse; hence the curvature of the ellipse is least, at the vertex of the conjugate axis. The curvature of the other two cur...