Download A Short Treatise on the Principles of the Differential and Integral Calculus Book in PDF, ePub and Kindle
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1829 edition. Excerpt: ... the same as that of-S, according to what was just observed, If the spiral change from concave to convex, 5? must change its sign, or at the point of inflexion-T-=0 or= oe. Hence the values of r, which gives either of these conditions, will shew the point of inflexion. ON THE DETERMINATION OF THE EX-PRESSION FOR THE RADIUS OF CUR-VATURE IN POLAR CURVES. The expression for the radius of curvature, referred to rectangular coordinates, assuming the positive sign for y, is That this value of y may be expressed in terms of the polar variables, we must eliminate the differential coefficients which enter into the formula by means of the following equations, x-r cos. 8, y--r sin. 8; which, being differentiated, and the results divided the one by the other, we shall obtain dy _ dr sin. 6 + r cos. 8 d8 dx dr cos. 6--r sin. 8 d8' and, representing the two terms of this fraction by m and n, we shall have m = dr sin. 8 + r cos. 8 d8, n--dr cos. 8--r sin. 8 d8. and consequently dy_m dx n dy2 _ m', dx2 n2 ' by means of which last equation we find for the numerator of the value of y, 3 and raising each term of this fraction to the power-, 3 and observing that the power-of n2 is n3, we have and dividing the first side of this equation by dx, and the second by, which is equivalent to dx, we shall have d'y _ ndm--mdu di.2 n1 By means of these values given by the two last equations, the expression for the radius of curvature becomes, (m2 + n2f y-ndm-mdn and we have now only to transform this equation into a function of 6 and r; for which purpose we must determine first the value of n2 + m2, by adding the squares of the value of m and n, and reducing by means of the equation sin.'9 + cos.'9 = I t Wfteft we shall find n2 + m2 = dr2 + r dS2. To obtain...