## Researches in the Calculus of Variations, Principally on the Theory of Discontinuous Solutions: An Essay to which the Adams Prize was Awarded in the University of Cambridge in 1871 |

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action admissible already angle apply assigned axis becomes boundary Calculus of Variations catenary Chapter circle condition consider consists constant continuous corresponding course curve cycloid denote determine diagram differential direction discontinuous solution discussion draw drawn equal equation examine example expression fact fixed points former given given length given points gives greater Hence hold horizontal imposed inclined increases infinite integral sign latter least less limits maximum method minimum namely necessarily negative obtain obvious origin parabola particular pass path portion positive possible preceding present problem proceed proposed quantity radius range reduces relation remark resistance respect result satisfied second order shew sin² solid solution stands straight line suppose surface taken tangents term tion touch usual vanish volume whole zero

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Page 163 - Is there any stationary solution for the brachistochrone when angular points are allowed ? 2. To find the form of a solid which experiences a minimum resistance when it moves through a fluid in the direction of the axis of revolution (Todhunter, Researches p.

Page 262 - An important remark must be made with respect to relative maxima and minima which, so far as I know, is not to be found in treatises on the subject.

Page 156 - A particle is projected from a given point with a given velocity and is acted on by a given force to a fixed point.

Page 156 - ... a fixed point by a force varying inversely as the square of the distance : determine the path of minimum action to a second fixed point.

Page 247 - Let 6 be the angle which the tangent to the curve makes with the ^ axis.

Page 134 - Prove that an arc of a circle from the lowest point which does not exceed 60° is a curve of quicker descent than any other curve which can be drawn within the same arc ; and...

Page 27 - Now x — - is the abscissa of the point of intersection of the tangent to the curve at the point (x, y) and the axis of x ; we will put f for this abscissa.

Page 76 - If p be the radius of curvature at any point of the concave portion we have .y=y Thus at the points P and Q since the curve touches the straight lines we have 22/! + £. = 2/t. 3a p a