# A Treatise on the Differential Calculus: And Its Applications to Algebra and Geometry, Founded on the Method of Infinitesimals

University Press, 1852 - Differential calculus - 540 pages

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Page 429 - but varies for different media; that is, the sine of the angle of incidence bears a constant ratio to the sine of the angle of refraction. The envelope of all the refracted rays is called the caustic by refraction of the given surface*.
Page 283 - cycloid is the curve traced out by a point in the circumference of a circle, as the circle rolls along a fixed straight line. (a) Let the given straight line (fig. 39) be taken as the axis of x, and the radius of the
Page 276 - and y have already preoccupied the two directions at right angles to each other in the plane of the paper, which is (and conveniently so) called the plane of reference, we must seek for some other course by which a line,
Page 303 - when x = a, a line parallel to the axis of y, at a distance a from it, is an asymptote ; and if x = oo , when y = b, a line parallel to the axis of x, at a distance
Page 44 - (fig. 4) be the circle, of which let the radius be a; take o the centre, and let the angle AOC be the rath part of four right angles, and AC be the side of a regular polygon of n sides inscribed in the circle ; and make
Page 335 - it is plain that the curve is symmetrical with respect to the axis of x, and since the curve passes through the origin, the
Page 283 - a curve, such that FT, the length of the tangent intercepted between the point of contact and the axis of x, is always equal to OA, then the locus of P is the equitangential curve. Let OM = x, MP = y, OA = PT = a; then the definition of the curve above given leads, as will be seen in the next Chapter, to an equation of the form
Page 246 - Ex. 7. To find a point within a triangle, such that the sum of the lines drawn from it to the angular points may be a minimum.
Page 390 - the length of the evolute is equal to the difference of the radii of curvature of the involute corresponding to its two extremities. Of this we subjoin some examples
Page 358 - the equation represents an ellipse, parabola, or hyperbola, according as e is less than, equal to, or greater than unity. Hence the equation to the parabola