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An elementary treatise on solid geometry - download pdf or read online

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By Smith, Charles, 1844-1916

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P, Q , t , P", S" Q", E", , {P"Q"R"S"}\ {P" Q" QR R" DEF. Two systems of planes, each of which has 37. a common line of intersection, are said to be homographic when every four constituents of the one, and the correspond ing four constituents of the other, have equal cross ratios. An equivalent definition [see Conies, Art. 323] is the two systems of planes, each of which has a following common line of intersection, are said to be homographic which are so connected that to each plane of the one system corresponds one plane, and only one, of the other.

Ex. 1. The locus of the centre of all plane sections of a conicoid which pass through a fixed point is a conicoid. The equation /, </, of the locus is (f-x)-^- + (y-y)-- +(h-z) j- = where > h are the co-ordinates of the fixed point. V Ex. 2. The locus straight line. of the centre of parallel sections of a conicoid is a DIAMETRAL PLANES. 44 The lx whose centre section + my + nzQ is (a, dF dF dF ~da _ dg ~ dy m~~ I Hence the locus is _I ^ dx m ~~ I The parallel the given to plane 11 the straight line whose equations are 1 planes 7) is /3, if i??

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


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