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

By Smith, Charles, 1844-1916

This e-book was once initially released sooner than 1923, and represents a replica of an enormous ancient paintings, preserving an identical layout because the unique paintings. whereas a few publishers have opted to follow OCR (optical personality reputation) know-how to the method, we think this ends up in sub-optimal effects (frequent typographical error, unusual characters and complicated formatting) and doesn't accurately guard the old personality of the unique artifact. We think this paintings is culturally very important in its unique archival shape. whereas we try to effectively fresh and digitally increase the unique paintings, there are sometimes situations the place imperfections corresponding to blurred or lacking pages, terrible photographs or errant marks can have been brought because of both the standard of the unique paintings or the scanning method itself. regardless of those occasional imperfections, we now have introduced it again into print as a part of our ongoing international e-book upkeep dedication, offering buyers with entry to the very best old reprints. We enjoy your knowing of those occasional imperfections, and in actual fact desire you take pleasure in seeing the ebook in a layout as shut as attainable to that meant via the unique writer.

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An elementary treatise on solid geometry

This ebook was once initially released sooner than 1923, and represents a duplicate of an incredible ancient paintings, holding an analogous layout because the unique paintings. whereas a few publishers have opted to practice OCR (optical personality acceptance) expertise to the method, we think this ends up in sub-optimal effects (frequent typographical mistakes, unusual characters and complicated formatting) and doesn't thoroughly safeguard the ancient personality of the unique artifact.

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Ex. 10. The locus of a point, which moves so that the sum of its distances from any number of fixed planes is constant, is a plane. Ex. 0. in a constant ratio, 21. The co-ordinates of any point on the line of intersection of two planes will satisfy the equation of each of the planes. Hence any two equations of the first degree represent a can find the equations of a straight line in straight line. their simplest form in the following manner. We be the straight line, pq its projection on the plane OZ.

50. 2 ax* + + cz* + 2fyz + 2^,7? + 2htvy -f 2w + 2vy + 2wz + d = 0, contains ten constants. But, since we may multiply or divide the equation by any constant quantity without altering the relation between as, y, and z which it indicates, there are really only nine constants which are fixed for any particular surface, viz. the nine ratios of the ten constants a, b, c, &c. to one another. surface of the second degree can therefore be made to satisfy nine conditions and no more. The nine % A conditions which a surface of the second degree can satisfy must be such that each gives rise to one relation among the constants, as, for instance, the condition of passing through a given point.

Z) = the equation found in the preceding Article will be zero. Two roots will be zero if I, m, n satisfy the relation If (a, 0, 7) be ; ,dF + l-T- da The line ^-^ = I t dF mjlz dp dF + n-r = V .................. < t dj 1$ = *^ n m w il] in that case be a contact being (a, /9, 7). tangent line to the surface, the point of If we eliminate /, m, n between the equations of the line, and the equation (i), we see that all the tangent lines lie in the plane whose equation is 39 THE POLAR PLANE.

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