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Page 69 - dx dx dy dy dz dz Differentiating the first of these equations with respect to x, the second with respect to y, and the third with respect to z ; putting x, y, and z each = 0, in the result, and making use of equations
Page 287 - Hence we conclude that the projections of the lines of curvature on the plane of the greatest and least axes of the ellipsoid, are the ellipses whose semiaxes, y, a, are connected by the equation (6). Thus the construction for describing them is as follows. Draw an ellipse, concentric with the ellipsoid, in the plane ca, with the lines
Page 290 - in (10), which will be a quadratic in v, we may determine two values of this parameter, which, substituted in (11), will give the equations to the hyperboloids of one sheet and of two sheets confocal with the ellipsoid, which cut it in the two lines of curvature passing through the given point
Page 70 - may be any point in one of the surfaces (a), and therefore each of these surfaces has its lines of curvature traced upon it by the surfaces of the series (a,), (« 2 ). Similarly it may be shewn that each surface, (a t ) has its lines of curvature traced by
Page 287 - below how f, g, h may be expressed symmetrically by two arbitrary constants, one of which is irrelevant, as it enters as a factor in the integral. From the equations (4), as from the ordinary forms, the properties of the projections of the lines of curvature may be readily deduced. Thus, taking the second, and substituting
Page 288 - that the projection of any line of curvature on the plane of the greatest and mean, or of the mean and least axes, does not meet its consecutive. It may be remarked with respect to equations (4), that any one of them may be deduced from any other, by combining it with the equation u + v
Page 233 - This will be ensured if the product x'y vanishes for every point in the plane x'y', and for every position of this plane passing through the principal axis OX' ; a definition equivalent to the one in which a principal axis is defined as an axis which is perpendicular to its diametral plane.
Page 235 - for P in the first member of (6), the first and second values will have contrary signs, and so will the third and fourth. Hence the cubic has a real root between a, /3, and another between /3, 7, and its remaining root must therefore also be real, and between oo and a, or between 7 and
Page 289 - that the lines of curvature of any surface of the second order with a centre, are its intersections with confocal surfaces of the second order. Since this property is independent of the centre, it follows that it must also be true for the case of a surface without a centre.
Page 40 - from it, except in such extreme cases as those in which the lines of curvature of the given series are indeterminate. In the method of curvilinear co-ordinates, as proposed by Lamé, the position of any point in space is determined by the three conjugate surfaces of the system (1) which intersect in the point. Thus, if any point