A Treatise on the Differential and Integral Calculus: And on the Calculus of Variations |
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angle apply asymptote axis b²x² C₁ circle co-ordinates constant cosec coversin d²u d2u d2u d2u dx2 d2z d2z d³y determine differential coefficients dq dx du du dx dx dx dy dz dx dz dx2 dy2 dx³ dxdy dy dy dx dz dx dz dy dz dz equal exponent expression F₁x formula fraction function given curve Hence imaginary increment h independent variable indeterminate form intersection lines of curvature logarithmic spiral logarithms maximum or minimum negative numerator obtain P₁ parabola parameters perpendicular plane curve power of h Prop r₁ radius of curvature radius vector respect result sec²x second differential coefficient sin x substitution subtan surface tangent 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 166 - 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. Put VD...
Page 202 - 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 223 - For when we differentiate the functional equation, first with respect to x and then with respect to y, we obtain the two equations tf'fcy) =/'(*). xf (X y) =/'(j,); and so, eliminating/'(a;i/), xf'(x) — yf'(y).
Page 112 - ... the first with respect to x, and the second with respect to y. Similarly we put J2 du , d?u ~dx <Pu ~d72 d*u . ,. and — ; — = . „ , ; the first expression indi2...
Page vii - ... 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, ' Is that answer perfectly correct?
Page 128 - 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.