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abcd angular points axes axis Cambridge centre of gravity chord circle circumscribing condition cone conic conicoid constant corresponding cosC cosé curve deduced determinant diameter differential direction direction-cosines distance eliminate ellipse ellipsoid equal equation equiangular spiral evidently expression force function Geometry given Hence intersection line at infinity line of curvature linear mathematical middle points multiplied normal obtain P. G. Tait parabola parallel parallelopiped perpendicular polar plane potency PROP proposition quadrilateral Quaternion radical axis radii radius vector ratio rectangular hyperbola represent result right angles scalar second order shew sides similar Similarly ſº soluble sphere straight line substitution of G suppose surface symmetrical relation tangent plane tetrahedron theorem triangle of reference trilinear coordinates values variables whence William Allen Whitworth
Página 206 - To study the nature of the surface more closely, let us find the locus of the middle points of a system of parallel chords.
Página 16 - For if we consider two particles of matter at a certain distance apart, attracting each other under the power of gravity and free to approach, they will approach ; and when at only half the distance each will have had stored up in it, because of its inertia, a certain amount of mechanical force. This must be due to the force exerted ; and, if the conservation principle be true, must have consumed an equivalent proportion of the cause of attraction; and yet, according to the definition of gravity,...
Página 15 - But from whence can this enormous increase of the power come ? If we say that it is the character of this force, and content ourselves with that as a sufficient answer, then it appears to me, we admit a creation of power, and that to an enormous amount ; yet by a change of condition, so small and simple, as to fail in leading the least instructed mind to think that it can be a sufficient cause : — we should admit a result which would equal the highest act our minds can appreciate of the working...
Página 15 - Assume two particles of matter, A and B, in free space, and a force in each or in both by which they gravitate towards each other, the force being unalterable for an unchanging distance, but varying inversely as the square of the distance when the latter varies.
Página 16 - B attract each other less because of increasing distance, then some other exertion of power either within or without them is proportionately growing up ; and again, that when their distance is diminished, as from 10 to 1, the power of attraction, now increased a hundredfold, has been produced out of some other form of power which has been equivalently reduced.
Página 16 - There is one wonderful condition of matter, perhaps its only true indication, namely inertia; but in relation to the ordinary definition of gravity, it only adds to the difficulty. For if we consider two particles of matter at a certain distance apart, attracting each other under the power of gravity and free to approach, they will approach ; and when at only half the distance each will have had stored up in it, because of its inertia, a certain amount of mechanical force. This must be due to the...
Página 40 - ... straight line DE, ie the poles of PQ and RG with respect to the hyperbola lie on the straight line DE. Therefore F is the pole of DE with respect to the rectangular hyperbola passing P, Q, R, and G; but P, Q, R, and G are the centres of the escribed and inscribed circles of the triangle DEF. Therefore if a rectangular hyperbola be so described that each angular point of a given triangle is the pole, with respect to it, of the opposite side, it will pass through the centres of the inscribed and...
Página 222 - To find the condition that the general equation of the second degree may represent two straight lines.
Página 156 - ... the straight lines joining the middle points of opposite edges of the tetrahedron. The edges of the tetrahedron are the diagonals of opposite faces of the parallelepiped.