The Differential and Integral Calculus: Containing Differentiation, Integration, Development, Series, Differential Equations, Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals,--with Applications to Algebra, Plane Geometry, Solid Geometry, and Mechanics. Also, Elementary Illustrations of the Differential and Integral Calculus
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Page 477 - ... connexion of the parts of the system one with another. Suppose perpendiculars drawn from the new positions of the points upon the directions of the forces acting at the points in their positions of equilibrium. The distance of any perpendicular from the original point of application of the corresponding force is called the virtual velocity of the point with respect to that force, and is estimated positive or negative, according as the perpendicular falls on the side of the point towards which...
Page 500 - ACD is a semicircular conducting loop of radius r with centre at O, the plane of the loop being in the plane of the paper. The loop is now made to rotate with a constant angular velocity co about an axis passing through O and perpendicular to the plane of the paper.
Page 565 - The history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, on account of one or more apparently valid cases in which such a method leads to erroneous results. Such cases should indeed teach caution, but not rejection; if the latter had been preferred to the former, negative quantities, and still more their square roots, would have been an effectual bar to the progress of algebra . . . and those immense fields of analysis over which even the...
Page 524 - Ex. 36. A particle on the concave surface of a sphere is repelled from the lowest point by a force which varies inversely as the square of the distance to the particle. Find the position of rest.
Page 399 - This plane, which passes through the centre of the sphere and is perpendicular to the chords it bisects, is called the diametral plane of the sphere for chords parallel to b.
Page 131 - English ; a maximum is not necessarily the greatest possible value of a function, nor a minimum the least. The greatest value of the function is the greatest of all its maxima, and the least value is the least of all the minima. A maximum may even be less than a minimum ; or the value of a function where its increase stops in one state may be less than that where its decrease stops in another state.
Page 109 - An interesting example for following out the processes of the engine would be such a form as x" dx or any other cases of integration by successive reductions, where an integral which contains an operation repeated n times can be made to depend upon another which contains the same n — 1 or n — 2 times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined. The methods in Arbogast's Calcul des Derivations are peculiarly fitted for the...
Page 570 - OA = -m, 0£ = + m, (fig. 33), we find the areas PAOY and QOBY both positive and infinite, which agrees with all our notions derived from the theory of curves. Again, if we attempt to find the area PYQB by summing PA 0 Y and YOQB, we find an infinite and positive result, which still is strictly intelligible.
Page 629 - In fact it has become usual to regard it as a method of oaaooa suggestion of new integrals to be verified by other methods rather than as a mode of investigation. For instance, De Morgan remarks : " It is a matter of some difficulty to say how far this practice may be carried, it being most certain that there is an extensive class of cases in which it is allowable, and as extensive a class in which either the transformation, or neglect of some essential modification incident to the manner of doing...