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 duplicate of a massive historic paintings, holding an analogous layout because the unique paintings. whereas a few publishers have opted to observe 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 competently look after the old personality of the unique artifact. We think this paintings is culturally vital in its unique archival shape. whereas we attempt to safely fresh and digitally improve the unique paintings, there are sometimes circumstances the place imperfections similar to blurred or lacking pages, bad photos or errant marks could have been brought because of both the standard of the unique paintings or the scanning procedure itself. regardless of those occasional imperfections, we have now introduced it again into print as a part of our ongoing worldwide ebook upkeep dedication, supplying buyers with entry to the absolute best ancient reprints. We have fun with your figuring out of those occasional imperfections, and clearly desire you take pleasure in seeing the e-book in a layout as shut as attainable to that meant by means of the unique writer.

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An elementary treatise on solid geometry

This e-book used to be initially released ahead of 1923, and represents a replica of an incredible historic paintings, retaining an identical structure because the unique paintings. whereas a few publishers have opted to practice OCR (optical personality popularity) know-how to the method, we think this ends up in sub-optimal effects (frequent typographical blunders, unusual characters and complicated formatting) and doesn't thoroughly safeguard the ancient personality of the unique artifact.

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Extra resources for An elementary treatise on solid geometry

Example text

I, elimination of I, 2 1, COS//, cosX, 0, o, therefore sin v cos 6= 1 COS 1 cos 2 X cos fju , cos X , Art 41, COS V 1 , COS /IP , cos 2 Hence the volume required 2 2 cos cos X ^ abc V (1 /-t , i , COS /JL COS \ , X o costf 1, cos e , COS V = , cosy, 2 (i) m, n from the above equations gives COS V 1, 0. 1 , 2 /-t. cos v 2 cos v -f +2 2 cos cos X X cos cos //, /u- cos cos v). TRANSFORMATION OF CO-ORDINATES. 44. To change the origin of co-ordinates without changing the direction of the axes. referred Let/, g, h be the co-ordinates of the new origin Let be any point whose co-ordinates to the original axes.

7) be (a, /3, 15 LINE. any point P on the P A and Draw through planes parallel to the co-ordinate planes so as to make a parallelepiped, and let AL, LM, be edges of this parallelepiped parallel to the axes of x, y, z on the axis is the Then projection of respectively. MP AP AL of#; therefore , x We a = lr, or X ~ have similarly y i r, m Hence the equations and OL = - zy = - n r. of the line are x OL _y I ft m _z 7_ n * Ex. 1. To find in a symmetrical form the equations of the line of inter section of the planes 5x - 4y = 1, By - 5z = 2.

Perpendicular to the line -4z. Find the angle between the 3. = 1 = and | f 1 U o = -^-. = 4 ^4ns. e 5 . 4. Shew that the lines 3x + 2y + z-5 = Q = x + y-2z-3, and 8x - 4y - 4z = = Ix + 1 Oy - 82 are at right angles. Ex. 5. Find the acute angle between the lines whose direction-cosines are Ex. 6. Shew that the straight lines whose direction-cosines are given by the equations 2l + 2m-n = Q, and mn + nl + lm = Q are at right angles. 2 Eliminating I, we have 2mn -(m + n) (2m -n) = 0, or 2in Hence, if the direction-cosines of the two lines be l^ m v n^ and have -1 - -|.

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