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

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A. S. Barnes, 1855 - 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 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.

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