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 angle asymptotes ax² axis b₁ centre co-ordinates conic conjugate point constant corresponding cubic cusp cycloid d²u denote derived functions determined differential coefficient double point du du dV dv dx dx dx dy dx dx² dxdy dy dy dy dz easily seen Eliminate ellipse envelope Euler's Theorem evidently evolute EXAMPLES expansion expression finite formula fraction geometrical Hence homogeneous function hypocycloid increment independent variable intersection inverse loga maxima and minima method minimum value negative nth degree origin osculating osculating circle parabola perpendicular point of contact point of inflexion positive prove quadratic radii radius of curvature reciprocal polar represented roots Salmon's second order substituting suppose tangent Taylor's Theorem Theorem tion triangle true value u₁ vanish y₁ zero аф
Popular passages
Page 173 - B — 0 = 60° ; and therefore the triangle is equilateral. 2. Find a point such that the sum of the squares of the perpendiculars drawn from it to the sides of a given triangle shall be a minimum. Let x, y...
Page 195 - ... divided at its point of contact into segments which are to each •other in a constant ratio. 6. Find the equation of the tangent at any point to the hypocycloid, «5 + yt - a* ; and prove that the portion of the tangent intercepted between the axes is of constant length.
Page 284 - 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 111 - ... cos a = cos b cos с + sin b sin с cos A ; (2) cos b = cos a cos с + sin a sin с cos в ; ^ A. (3) cos с = cos a cos b + sin a sin b cos C.
Page 173 - 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 279 - The curve is symmetrical with respect to the axis of x, and has two infinite branches ; the origin is a double cusp. The shape of the curve is exhibited in the figure annexed.
Page 185 - Polar Co-ordinates. — The position of any point in a plane is determined when its distance from a fixed point called a pole, and the angle which that distance makes with a fixed line, are known ; these are called the polar co-ordinates of the point, and are usually denoted by the letters r and 0. The fixed line is called the prime vector, and r is called the radius vector of the point. The equation of a curve referred to polar co-ordinates is generally written in one or other of the forms, r =/(0),...
Page 196 - 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 234 - One angle of a triangle is fixed in position, find the envelope of the opposite side when the area is given. 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. 5. From a point P on the hypothenuse of a right-angled triangle, perpendiculars PM, PJV are drawn to the sides ; find the envelope of the line MN.
Page 255 - ... 237. Evolutes and Involutes. — If the centre of curvature for each point on a curve be taken, we get a new curve called the evolute of the original one. Also, the original curve, when considered with respect to its evolute, is called an involute. To investigate the connexion between these curves, let...