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

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A.S. Barnes & Company, 1856 - 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 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 231 - Now differentiating both (1) and (3) first with respect to x, and then with respect to y, we...
Page 168 - A cycloid is the curve generated by the motion of a point on the circumference of a circle rolled in a plane along a straight line. If the circle is rolled on the outside of another circle, the curve generated is called an "epicycloid"; if rolled on the inside, it is called a "hypocycloid.
Page 130 - The general form to which every complete equation of the second degree may be reduced, is z2+2pz=g ; in which 2p and q may be either both positive or both negative, or one positive and the other negative. Completing the square, we have Now, the first member is equal to (z+p)2, and if, for the sake of simplicity, we assume g+p2=?2.
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 - B are unknown z2 + bx xx + b constants whose values are to be determined by the condition that this assumed equality shall be verified. Reducing the terms of the second member to a common denominator, we have...
Page 365 - Homogeneous Exact 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 differential expression is said to be homogeneous when the sum of the exponents of the variables is the same in the coefficient of every term. Thus xdy -\- ydx, x2zdx + xzWx — xyzdy, and -y are homogeneous differentials.
Page 107 - Similarly, z2 = x, x2 = x, &c., and, therefore, the parts are all equal. 16. To determine the number of equal parts into which a given number a must be divided, so that their continued product may be a maximum. Let x = required number of parts ; then - = value of one part. aaa , (a\* . • . - X - X - &c.
Page 356 - ... of the wedges which constitute the entire volume. 118. 1. The hemisphere with radius equal to a. Here the limits of the integrations are r = 0 and r — a, — 0 and a = ~, v = 0 and v = 2*.

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