By Smith, Charles, 1844-1916

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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.

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

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