A Treatise on the Differential and Integral Calculus: And on the Calculus of Variations |
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angle asymptote axis b²x² circle co-ordinates constant cosec coversin d²u d2u d2u d2u dx2 d²y d2z d2z d³y determine differential coefficients differential equation dx dx dx dy dx dz dx₁ dx² dx2 dy2 dx³ dxdy dy dy dx dy dz dz dx dz dy dz dz dz₁ exponent expression F₁a F₁x formula fraction function generatrix given curve Hence imaginary increment h independent variable indeterminate form intersection lines of curvature logarithmic spiral logarithms maximum or minimum negative normal numerator obtain P₁ parabola parameters perpendicular plane curve power of h Prop R₁ radius of curvature respect result sec²x second differential coefficient substitution subtan surface surfaces of revolution Taylor's Theorem u₁ x₁ y₁ Φια
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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.
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 210 - Hence y = ±(x — - b) 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 129 - ... h and y by y ± k, when h and k may take any values between zero and certain small but finite quantities ; and u is said to be a minimum when its value is less than all other values determined by the conditions above described. 97. Prop. Having given u = F(x, y), when x and y are independent variables, to determine the values of a; and y which shall render ua maximum or minimum.
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 368 - X. Containing three variables (39) 178 Containing more than three variables . . . (40) 178 XI. Containing three variables . . . . . . (41) 178 Containing more than three variables . . . (42) 179 (1) Of or reducible to the form Xdx + Ydy = 0, where X is a function of x alone and У is a function of y alone. Integrate each term separately, and write the sum of their integrals equal to an arbitrary constant. (2) M and N homogeneous functions of x and y of the ~same degree. Introduce in place of y the...
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 iii - ... the mathematical chair was Edward H. Courtenay, from 1842 to 1853. He was the first regular occupant of this chair who was educated in this country. He was born in Baltimore, in 1803. After having been examined for admission to the US Military Academy at West Point, in 1818, the examiner remarked : " A boy from Baltimore, of spare frame, light complexion, and light hair, would certainly take the first place in his class.