An Elementary Treatise on the Differential Calculus: With Applications and Numerous Examples

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Macmillan, 1892 - 521 páginas
 

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Página 148 - ... equation of the locus of the foot of the perpendicular from the pole on to any tangent to the curve.
Página 8 - The opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles, with converse.
Página 417 - An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water.
Página 26 - When a particle is in motion in a given manner the space described is a function of the time of describing it. We may consider the time as an independent variable, and the space described in that time as the dependent variable. The rate of change of position of the particle is called its velocity. If uniform the velocity is measured by the space described in one second ; if variable, the velocity at any instant is measured by the space which would be described in one second if, for that second, the...
Página 18 - Tangents are drawn to a circular arc at its middle point and its extremities; A is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and A' the area of that formed by the three tangents.
Página 28 - Prove that if a particle move so that the space described is proportional to the square of the time of description, the velocity will be proportional to the time, and the rate of increase of the velocity will be constant.
Página 417 - ... 40. Find the mass of an elliptic plate of semi-axes a and 6, the density varying directly as the distance from the centre and also as the distances from the principal axes. 41. From a fixed point A on the circumference of a circle of radius a, the perpendicular AY is let fall on the tangent at P. Prove that the greatest area APY can have is -~— a 2 . 42.
Página 408 - A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, required the height and breadth of the window when the quantity of light admitted is a maximum. The radius of the semicircle must equal the height of the rectangle.
Página 212 - ... is called a point of contrary flexure, or a point of inflexion.
Página 2 - As not everyone can cope unaided with mathematical technicalities, the following may be useful : ' When one quantity depends upon another or upon a system of others, so that it assumes a definite value when a system of definite values is given to the others, it is called a function of those others...

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