The Doctrine of Germs, Or, The Integration of Certain Partial Differential Equations which Occur in Mathematical Physics

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Deighton, Bell, and Company, 1881 - Differential equations, Partial - 108 pages
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Page 48 - The locus of the middle points of any system of parallel chords of a parabola is a straight line parallel to the axis...
Page 80 - SECOND FOCUS AND DIRECTRIX. PROPOSITION III. 35. Every central conic has a second focus and directrix ; and the sum of the focal distances of any point on the curve in the case of the ellipse^ or the difference of the same in the case of the hyperbola., is constant and equal to the transverse axis. The existence of a second focus and directrix has been proved in Art. 14, Cor. 3 ; but it may also be deduced from the relation PN* : CA*~Pn*= Cff : CI?, * Upon this subject, see Scholium C.
Page 212 - P, which moves so that its distance from a fixed point is always in a constant ratio to its perpendicular distance from a fixed straight line, is called a Conic Section.
Page xxxvii - ... solidity of the pyramid will still be equal to one third of the product of the base multiplied by the altitude, whatever be the number of sides of the polygon which forms its base- • hence, the solidity of the cone is equal to one third of the product of its base multiplied by its altitude.
Page xlv - Hultsch). circle : the term Cone is used specially of the finite portion of the superfices between the vertex and the circle or base. The Axis is the line from the vertex to the centre of the base. The plane through the axis at right angles* to the base cuts the cone and its base in a triangle, which is called " the triangle through the axis ;" and every chord of the cone at right angles to the plane of this triangle is bisected by it (prop. 5). Any plane at right angles to the plane of the triangle...
Page 42 - The triangle whose angular points are the focus of a conic and the intersections of the tangent and the diameter at any point with the axis and the directrix respectively has its orthocentre at the point in which the tangent meets the directrix. 57. Given the focus and the directrix of a conic, shew that the polar of a given point with respect to it passes through a fixed point. 58. If the polar of a point 0 with respect to a conic intersect a conic having the same focus and directrix in P, and if...
Page 211 - ... extremity of the shadow of a vertical gnomon erected on a horizontal plane, on a given day and in a given latitude. 564. The centre of a sphere moves in a room a vertical plane which is equidistant from two candles of the same height from the floor: determine its locus if the shadows upon the coiling be always in contact. 565. If a point move in a plane in such a way that the sum or difference of its distances from two fixed points, one of which lies in the plane and the other without it, is...
Page 225 - Appendix (sec above p. 219, note). 607. The radius of curvature at any point of a parabola is double of the portion of the normal intercepted between the curve and the directrix. 608. At any point of a parabola the intercept made by the circle of curvature upon the axis is a third proportional to the latus rectum and the parameter of the diameter to the point.
Page 326 - OI/On, and On is constant and na fixed point. 2. Another proof is given as a problem in The Ancient and Modern Geometry of Conies, page 122 (1881), thus, " 279. If PQ be a chord of a conic which subtends a right angle at a given point...
Page 194 - A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.

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