Oxford, Cambridge and Dublin Messenger of Mathematics, Volume 4Macmillan and Company, 1868 - Mathematics |
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Common terms and phrases
a₁ acceleration acnode algebraical anapole asymptotic axes axis B₁ B₂ blank lots C₁ centre of gravity coefficient coincide cone confocal conic section constant coordinates cos² cose curvature curve denote determine double points ellipse ellipsoid equal equation expression f+gh finite fraction function g+hf geometrical given h+fg Hence horizontal hyperbola hyperbolic paraboloid hyperboloid infinity initial integral intersect limit line joining locus meet moment of inertia motion multiplying nine-point circle obtained origin orthotomic P₁ parabola parallel parcels perpendicular plane point of contact position ratio respectively right angle sides Similarly sin² singular solution straight line surface tangent tension THEOREM things tion triangle ABC trilinear trilinear coordinates unity vanish vertex vertical x₁ zero
Popular passages
Page 97 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 77 - It is also known* that the moment of inertia about any axis whatever, is equal to the moment about a parallel axis through the centre of gravity, together with the moment which the whole mass, if collected at the centre of gravity, would have about the original axis.
Page 4 - An elliptic lamina is supported, with its plane vertical and transverse axis horizontal, by two weightless pins passing through its foci. If one of the pins be released, determine the eccentricity of the ellipse in order that the pressure on the other may be initially unaltered.
Page 79 - P. In a manner precisely similar, it may be shewn that if we draw through P an hyperboloid of two sheets confocal to the ellipsoid of gyration, the value of X for its tangent plane at P is a minimum; and therefore that the normal to this surface at P is the principal axis of greatest moment at P. This being so, we know that the remaining principal axis is perpendicular to these two, and is therefore normal to the confocal hyperboloid of one sheet which passes through P. We have proved then that the...
Page 18 - T be the point of concourse of the common tangents of an ellipse and its circle of curvature at P...
Page 193 - In general, the equation above gives the number of ways in which N things can be arranged in groups, the numbers Ni, N2, etc., representing the number in each group. In the Maxwell-Boltzmann statistics the "things" to be arranged are the phase points, the number of "groups" equals the number of cells in phase space, and the number of ways of arranging the "things...
Page 150 - Let a, /S, 7.../c denote the n things, and let N represent the number of ways in which they can be permuted when unrestricted by any condition. Also let (A) express the condition that a is in its original position, and (a) the condition that a is out of its original position. Let (B} and (b) denote the corresponding conditions with respect to /3, and so on.
Page 193 - The number of ways in which n different things can be arranged in r indifferent groups (blank lots being inadmissible] is [n.
Page 79 - If the two points of contact coincide, then at that point two principal axes are perpendicular to the line, which is therefore itself a principal axis. The condition, therefore, that an axis may be a principal axis at some point of its length, is that the two points of contact of confocal surfaces touching it must coincide ; which is obvious, for in that case the line is normal to the third surface passing through the common point of contact. * Salmon's Geometry of Tflree Dimensions, p.
Page 152 - The total number of partitions of n different things (ie the number of ways in which n different things can be distributed into 1, 2, 3, ... or n indifferent parcels) is the coefficient of x in the expansion of |n.ee —1 Solution by the REV.