A Treatise on the Differential and Integral Calculus: And on the Calculus of Variations

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A. S. Barnes & Company, 1857 - Calculus - 501 pages
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Page iv - An Elementary Treatise on Mechanics. Translated from the French of M. Boucharlat. With Additions and emendations, designed to adapt it to the use of the Cadets of the US Military Academy.
Page 168 - When y = 0, r = ao , and when y = ± oo , r = oo . 5. The cycloid, or curve generated by the motion of a point on the circumference of a circle, while the circle rolls on a straight line. Let the radius of the generating circle = a. Place the origin at V, the vertex of the cycloid.
Page 210 - Hence y — ±(x — - 6) is the equation of two straight lines, which are asymptotes to the curve, and are inclined to the axis of x at angles of 45° and 135° respectively. If we combine this equation of these asymptotes with that of the curve, we shall find that each of the asymptotes intersects that branch of the curve which lies on the right of the axis of y. Forming the value of...
Page v - Mr. Courtenay was a mathematician of noble gifts and a great teacher. " His mind was quick, clear, accurate, and discriminating in its apprehensions, rapid and certain in its reasoning processes, and far-reaching and profound in its general views. It was admirably adapted both to acquire and use knowledge.''t He was modest and unassuming in his manner, even to diffidence. He would never utter a harsh word to pupils or disparage their efforts. " His pleasant smile and kind voice, when he would say,...
Page 261 - Ex x2 + bx ~ x2 + bz x2 + bx~ z2 + bx Hence ax + c = Ax + Ab + Bx ; and since this condition is to be fulfilled without reference to the value of x, the principle of indeterminate coefficients will furnish the separate equations c = Ab, and a = A + B.
Page 129 - = - — . • . h = . 11 and a = 1 . 5 + . 11 = 1 . 61 16 nearly. And if we repeat the operation by putting x — 1 . 61, a nearer approximation will be obtained. CHAPTER X. • MAXIMA AND MINIMA FUNCTIONS OF TWO INDEPENDENT VARIABLES. 96. A function u of two independent variables x and y, is said to be a maximum when its value exceeds all those other values obtained by replacing x by x...
Page 363 - Differentials. 131. Although the methods of integration just explained apply to all exact differentials, yet another and simpler process can be used when the expression belongs to the class called homogeneous. A...
Page 344 - OF VOLUMES. 111. Prop To obtain a general formula for the volume generated by the revolution of a plane figure about a fixed axis. Let OX, the axis of x, be the axis of revolution, ABCF the generating area. Put and let y — Fx be the equation of the of 00, bounding curve AS.
Page 112 - Eq. (4.7), we may take partial derivatives of the first with respect to x and the second with respect to y and...
Page 368 - C, where C is an arbitrary constant. 139. Again, the integration can be effected whenever the separation of the variables is possible, that is, when the equation can be reduced to the form Xdx + Tdy = 0, whore X is a function of x only, and Y a function of y only.

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