Prof. Cayley on the Projection of the Ellipsoid. 



51 



the section; and in particular, if the section is central, that 

 is, if the plane thereof passes through the centre of the ellip- 

 soid, then the pole is the point at infinity on the conjugate 

 diameter; whence also if the eye be at an infinite distance, 

 so that the projection is a projection by parallel rays, then the 

 projection of the pole is the point at infinity on the projection of 

 the conjugate diameter; and therefore the common tangents of 

 the projections of the section and the contour section are in this 

 case parallel to the projection of the diameter conjugate to the 

 plane of the section. 



Suppose that the plane of the picture is parallel to a principal 

 plane of the ellipsoid, aud that the projection is by parallel rays ; 

 then if OA, OB, OC are the projections of the semiaxes (OA, 

 OC will be at right angles to each other if the plane parallel to 

 the plane of the picture is that of xz), the projections of the prin- 

 cipal sections are the ellipses having for conjugate semidiameters 

 OB, OC ; OC, OA ; OA, OB respectively. Hence to the ellipse 

 OB, OC drawing the two tangents which are parallel to OA, to 

 the ellipse OC, OA the two tangents which are parallel to OB, 

 and to the ellipse OA, OB the two tangents which are parallel 

 to OC, we have on each of these ellipses the two points which 

 are the points of contact therewith of the ellipse which is the 

 projection of the contour section, or apparent contour of the 

 ellipsoid ; that is, we know six points, and at each of these points 

 the tangent, of the last-mentioned ellipse; and the ellipse in 

 question, or apparent contour of the ellipsoid, can thus be traced 

 by hand accurately enough for ordinary purposes. 



In connexion with with what precedes, I may notice a conve- 

 nient construction for the pro- 

 jection of a circle. Suppose 

 that we have given the pro- 

 jection of the circumscribed 

 square A B C D ; then if we 

 know the projection of one of 

 the points M, N, P, Q, say of 

 the point M, the projections 

 of all the points and lines of 

 the figure can be obtained gra- 

 phically by the ruler only with 

 the utmost facility; that is, 

 in the ellipse which is the 

 projection of the circle we 

 have eight points, and the tan- 

 gent at each of them, and the ellipse may then be drawn by 

 hand. And to find the projection of the point M, it is only 

 necessary to remark that in the figure the anharmonic ratio 



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