An Elementary Treatise on the Differential Calculus: Containing the Theory of Plane Curves, with Numerous Examples |
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Common terms and phrases
accordingly algebraic angle asymptotes ax² axis becomes infinitely centre circle co-ordinates conic conjugate point constant corresponding cubic cusp d²u d³u denote derived function determined dF dF differential coefficient double point dV dV dx dx dy dx dy dx dx² dxdy dy dy dy dz easily seen ellipse equation Euler's Theorem evidently evolute EXAMPLES expansion expression finite fraction geometrical given curve Hence homogeneous function imaginary increment independent variable infinitely small intersection inverse limit Maxima and Minima minimum value negative nth degree origin osculating osculating circle parabola perpendicular point of contact point of inflexion positive prove quadratic quantity radii radius of curvature reciprocal polar represented roots Salmon's substituting subtangent suppose tangent Taylor's Theorem triangle U₁ vanish y₁ zero аф
Popular passages
Page 217 - Cartesian oval is of the form r + kr' = a, where r and r' are the distances of any point on the curve from two fixed points, and a, k are constants.
Page 282 - If two curves have a contact of the nth order, no curve having with either of them a contact of a lower order can fall between the curves near their point of contact. (2). Two curves which have a contact of the nth order at a point are infinitely closer to one another near that point than two curves having a contact of an order lower than the nth.
Page 255 - ... 3. Find the envelope of a line when the sum of the squares of the perpendiculars on it from two given points is constant. 4. Find the envelope of a right line, when the rectangle under the perpendiculars from two given points is constant.
Page 306 - This curve is the path described by a point on the circumference of a circle, which is supposed to roll upon a fixed right line.
Page 185 - Find the locus of a point such that the sum of the squares of its distances from two fixed points shall be equivalent to the square of the distance between the fixed points.
Page 216 - In the curve xn ->- y" = a", find the length of the perpendicular drawn from the origin to the tangent at any point, and find also the intercept made by the axes on the tangent. 8. Find the length of the normal at any point in the catenary V* — ef - -*-\ , = -\ ec + e « J. An2. — . 9. Find an expression for the polar subtangent to the curve r (e
Page 216 - In the same curve prove that the portion of the tangent intercepted between the axes is divided at its point of contact into segments which are to each other in a constant ratio.
Page 173 - Tan"1« . tan" (a — x) is a maximum, when tan (a — 2x)= tan a ? 1 8. Prove that • is a minimum when x = e. log» 19. Given the vertical angle of a triangle and its area, find when its base is a minimum. 20. Given one angle A of a right-angled spherical triangle, find when the difference between the sides which contain it is a maximum. Here tan с cos A = tan b ; and, since с — Ь is a maximum, — = I.
Page 207 - Anti-Friction Pivot. The theoretical advantage of the adoption of the form in this case is that the vertical wear of the pivot and step is everywhere equal. INVERSE CURVES. DEF. If on any radius vector OP drawn from a fixed origin 0, a point P be taken such that the rectangle OP . OP...
Page 217 - Prove that the locus of the foot of the perpendicular from the pole on the tangent to an equiangular spiral is the same curve turned through an angle.