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By Smith, Charles, 1844-1916

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1) line of constant length has its extremities on two fixed straight lines; the locus of its middle point is an ellipse. take the axes of co-ordinates as in Art. 35, the equations of the lines z=-c. Let the co-ordinates of the and z = A shew that If we will be y = mx, c\ y= -mx, EXAMPLES. 33 any one of its possible positions be a^, y v z l and x y, z ) the co-ordinates of the middle point of the line. ( be the length of the line, we have extremities of the line in x Then, y< >2> 22 allcl lct l>o > if 21 But, since y 1 = mx l and 21 ^=c, and #2 = -wo;2 2 = 0, A Z 2 = -c, we have is the ellipse whose equations are = moves so as always line z2 -z 2 = 2c, and Hence the locus of the middle point (2) , to intersect three which are not all parallel to the same surface generated by the straight line.

Since one root of a cubic is always real, it follows that there is always one principal plane. ; Find the principal planes of the following surfaces : + 2zx~2xy + Wy* + &z*-8yz + 4z x* + y*-z * + 2yz < \/ (i) (ii) llx> Ans. Ans. 61. and (i) (ii) All parallel plane sections of a conicoid are similar similarly situated conies. Change the axes of co-ordinates one of the system plane of xy may be in such a way that the of parallel planes ; and let the equation of the surface be the general equation of the second degree.

Hence (P QRS] = [P Q R S }, we have 8"}={P[ which proves the proposition. 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.

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

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