This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 edition. Excerpt: ...in other respects be the same as for the hyperboloids of two sheets, and when the difference of potentials is given the density at any point of the surface will be proportional to the perpendicular from the centre on the tangent plane, and the whole quantity on the infinite sheet will be infinite. Limiting ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 edition. Excerpt: ...in other respects be the same as for the hyperboloids of two sheets, and when the difference of potentials is given the density at any point of the surface will be proportional to the perpendicular from the centre on the tangent plane, and the whole quantity on the infinite sheet will be infinite. Limiting Forms. (1) When /3 = 0 the surface is the part of the plane of xz between the two branches of the hyperbola whose equation is written above, (24). (2) When fi=F(k') the surface is the part of the plane of xy which is on the outside of the focal ellipse whose equation is (25) The Ellipsoids. For any given ellipsoid y is constant. If two ellipsoids, yJ and y2, be maintained at potentials V1 and V2, then, for any point y in the space between them, we have r.ny(yi-, (26) The surface-density at any point is, i5=5 a. (27) where p3 is the perpendicular from the centre on the tangent plane, and P3 is the product of the semi-axes. The whole charge of electricity on either surface is given by Q2 = clf =-ql, (28) ri ft and is finite. When y = F(k) the surface of the ellipsoid is at an infinite distance in all directions. If we make I2 = 0 and y2 = F(k), we find for the quantity of electricity on an ellipsoid maintained at potential V in an infinitely extended field, V Q = 1Wr" (29) The limiting form of the ellipsoids occurs when y = 0, in which case the surface is the part of the plane of xy within the focal ellipse, whose equation is written above, (25). The surface-density on either side of the elliptic plate whose equation is (25), and whose eccentricity is h, is V 1 1 (30) Particular Cases. 151. If c remains finite, while b and therefore h is diminished till it becomes ultimately zero, the system of surfaces becomes transformed in the following...
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