Differential Equations |
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Common terms and phrases
A₁ a²f a²ƒ arbitrary constants arbitrary function canonical form coefficients consequently const continuous function converges Crelle's Journal cusp-locus definite integral denote derived dx dx dx dy dy dx eliminating envelope equation dy equation is satisfied equation of type exact differential expression given Green's function Hence hypergeometric equation integral curves integrating factor interval Laplace's equation Legendre's equation method obtain orthogonal trajectories p-curves p-discriminant P₁ partial differential equation plane positive integer Prove quantities regarded relation second order singular solution solution of equation Solve the equations substitution supplementary conditions surface tangent theorem transformed u₁ variables write Y₁ zero θα дг ди др дф дх ду дь მა მე მყ
Popular passages
Page 291 - ... or the cone and the cylinder are kept steady, or, in other words, without rotation on their axes, the ball can roll along them in contact with both, so that the point of rolling contact between the ball and the cylinder shall traverse a straight line on the cylindric surface parallel necessarily to the axis of the cylinder — and so that, in the case of a...
Page ii - GENERAL EDITOR: P.. ABBOTT, BA, HEAD OF THE MATHEMATICAL DEPARTMENT, THE POLYTECHNIC, REGENT STREET, LONDON Now ready A Manual of Mechanical Drawing. By JH DALES, CE 3s net ; or in two parts, 2s net each.
Page 292 - ... be called the ball's longitudinal motion. Now for the integration of ydx \ the distance of the point of contact of the ball with the disk or cone from the centre of the disk or vertex of the cone in the ball's longitudinal motion is to represent y, while the angular space turned by the disk or cone from any initial position represents x ; and then the angular space turned by the cylinder will, when multiplied by a suitable constant numerical coefficient, express the integral in terms of...
Page 252 - Pn(#). both are interior to R, and one such that on any ordinate of the interval all the points between (x, P(x)) and (x, Q(x)) are also interior points of R. Consider now a closed region RI interior to R and containing the point...
Page 291 - ... or cone, when both considered as lines traced out in space fixed relatively to the framing of the whole instrument, will be two parallel straight lines, and that the line of motion of the ball's centre will be straight and parallel to them. For facilitating explanations, the motion of the centre of...
Page 293 - ... a movable point, moved by the hand of an operator, so as always to touch the curve on the surface of the cylinder, while the two cylinders are moved round. Two operators will be required, as one operator could not move the two points so as to fulfil this condition — at all events unless the motion were very slow. One of these points, by proper mechanism, gives an angular motion to the rotating disk equal to its own linear motion, the other gives a linear motion...
Page 251 - Any value x > £ will lie on one of the intervals z»_ixn, and the polygon will either be well-defined for all such values, or else there will be a constant /3 such that for every x in the interval £ ^ x...
Page 291 - The new principle consists primarily in the transmission of motion from a disk or cone to a cylinder by the intervention of a loose ball, which presses by its gravity on the disk and cylinder, or on the cone and cylinder, as the case may be, the pressure being sufficient to give the necessary frictional coherence at each point of rolling contact ; and the axis of the...
Page 253 - ... the slope of any side of any one of them is numerically less than m. It follows that each is certainly well defined and within the rectangle over an. interval £ ^ x ^ a\, where ai is the smaller of a and b/m.
Page 253 - Consider the function satisfying the differential equation «'- kv+l and having |uo| as its initial value* The value of u is never greater than that of v, since otherwise the difference u — v would vanish and have a positive or vanishing forward derivative at some point. At a point where u and v are equal, however, |«'| < fc|«| + I = kv + I = v', which is a contradiction.