A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Volumen 2
The University Press, 1854
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angle arbitrary constants arbitrary function axis calculus of variations coefficients complete integral consider contained convenient corresponding Crelle's Journal critical value deduced definite integral derived functions derived-functions determined differential calculus double integral dx _ dx dy dy dx element-function involves ellipse ellipsoid equa equal equicrescent exact differential expressed f dx factor finite formulae geodesic lines geometrical given curve given points Hence hereby inferior limit infinitesimal element-function infinitesimal elements integral becomes intersection investigate length limits of integration lines of curvature maxima and minima means method multiplying normal partial differential partial differential equation perpendicular plane curve problem properties quantities radius of curvature result right-hand member satisfied shewn similar similarly singular solution straight line substituting surface symbol Theorem tion undetermined vanish variables volume whence y-integration
Página 164 - ... is equal to the product of the length of the curve and the length of the path described by the centroid of the curve.
Página 284 - ... of any form and size. When stretched the rod becomes thinner, so that the several particles undergo lateral as well as longitudinal displacements. There is one fibre or line of particles which is undisturbed by the lateral contraction. Let this straight line, which we may regard as the central line, be taken as the axis of x, and let the origin be at the fixed extremity of the rod. We suppose that the stretching forces at the two ends are distributed over the extreme cross sections in such a...
Página 502 - Find the curves in which the perpendicular from the origin on the tangent is equal to the abscissa of the point of contact. [The circles r - 2a cos в.] 13.
Página 292 - F3 are proportional to the directioncosines of the normal to the surface at the point (x, y, z), that the points determined through these equations are such that lines joining them to the point (a, b, c) stand normal to the surface. If...
Página 510 - By making *' = 0, we find the co-ordinates of the point where the normal meets the plane of xy, also, the length of the normal, intercepted between the surface and the plane of wy, is РнoB.
Página 168 - ... and one of whose edges passes through the centre of the sphere. Find the area of the surface of the sphere intercepted by the cylinder. .Let the cylinder be perpendicular to the plane of xy ; then the equations of the cylinder and the sphere are respectively yz = ax — x2 and я2 -f y2 + z2 = a*.
Página 346 - R ' ( } where P, Q, and R are functions of x, y, and z.
Página 306 - It is not possible to have a series of geodesic lines passing through a point and touching a given line of curvature. In fact the words "originating at a point" should be omitted, as they are not required or used. On page 308 we read " it may be proved in the same way as the analogous theorem in plane geometry, that the geodesic radii vectores make equal angles with the curve of curvature.
Página 502 - A. + u for the rectification of a plane curve, where p is the perpendicular from any assumed point called the pole on a tangent to the curve, A the angle between this perpendicular and any fixed line drawn through the pole, u the portion of the tangent intercepted between the point of contact and the foot of this perpendicular. Let Q be the centre of curvature of the arc at A, AB a tangent at A...