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New PDF release: An elementary treatise on solid geometry

By Smith, Charles, 1844-1916

This e-book used to be initially released sooner than 1923, and represents a duplicate of a major old paintings, holding an analogous layout because the unique paintings. whereas a few publishers have opted to follow OCR (optical personality acceptance) expertise to the method, we think this results in sub-optimal effects (frequent typographical error, unusual characters and complicated formatting) and doesn't accurately shield the historic personality of the unique artifact. We think this paintings is culturally vital in its unique archival shape. whereas we try to effectively fresh and digitally improve the unique paintings, there are sometimes circumstances the place imperfections corresponding to blurred or lacking pages, bad images or errant marks could have been brought because of both the standard of the unique paintings or the scanning technique itself. regardless of those occasional imperfections, we've got introduced it again into print as a part of our ongoing international e-book protection dedication, offering shoppers with entry to the absolute best ancient reprints. We enjoy your knowing of those occasional imperfections, and in actual fact desire you get pleasure from seeing the e-book in a layout as shut as attainable to that meant by way of the unique writer.

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Extra resources for An elementary treatise on solid geometry

Example text

Dx dy dz 59. To find the locus of the middle points of parallel chords of a conicoid. dF +m dF t l-j- da -^5 d/3 + ft dF = -5- . 0. dF , c -i |- ax m dF -j dy \- n dF = -y- 0. dz Def. The locus of the middle points of a system of parallel chords of a conicoid is called the diametral plane. If the plane be perpendicular to the chords it bisects, it is called a principal plane. 60. To find the equations of the principal planes of a conicoid. The diametral plane are I, m, n 7 l dF ~j ax or, I + whose dF = O m dF -J- + n TJdz writing the equation in (ax or of the chords direction- cosines is v/ dy full, + hy +gz + u) + m (hx + by +fz + v) + n (gx +fy -f cz + w) 0, x (al + hm + gri) + y (hi + bm +fri) + z (gl +fm + en) + ul + vm 4- wn 0.

DF A m dF + n-j-^O ............. -,^ dft dy ... (i). In this case (a, ft, 7) will be the middle point of the chord. Hence an infinite number of chords of the conicoid have the point (a, ft, 7) for their middle point. dF Hence n between the equations . dF 7) is the centre of the conic in which .... (ii) meets the surface. This result should be compared with that obtained in Art. 52. Ex. 1. The locus of the centre of all plane sections of a conicoid which pass through a fixed point is a conicoid.

Shew that there are only two independent conditions, as is geometrically obvious, for if the planes have two points in common they must have a common line of intersection. It is easy to THE PLANE. Now if PL 13 be the perpendicular from P on the plane (i), and ON, ON the perpendiculars from the origin on the planes and (ii) respectively, then will (i) =p -p = lx + my -f nz p. of the perpendicular from any point on nz p = is obtained by substituting the Hence the length the plane lx + my + co-ordinates of the point in the expression lx If the equation of the plane be may be -f my -\-nz-p.

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An elementary treatise on solid geometry by Smith, Charles, 1844-1916

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