184 



Parallel tangents being drawn to an ellipse and its evolute, and 

 perpendiculars from the centre let fall upon them, the difference of 

 the squares of these perpendiculars is equal to the square of the semi- 

 diameter of the ellipse which coincides with the perpendiculars. 



I will proceed with a few other applications of the method. For 

 example, 



A surface of the second order touches seven given planes, to find 

 the locus of its centre. 



Let the tangential equation of the given surface be 



and let the twenty-one coordinates of the seven given planes be 

 I', v', f ; ?', v", t" ; r ", v'", '", &c. Substituting these values 

 successively in the preceding equation, we shall have seven linear 

 equations by which we may eliminate the six quantities a, a', a" ; 

 ft /3' |3". The resulting equation will also be linear, and of the form 



which is the equation of a plane. Now y, y t , and y,,, as may be 

 shown, are the protective coordinates of the centre of the surface. 

 Hence the centre of the surface moves along a plane. When there 

 are eight planes, we may then eliminate y or y,, and the two result- 

 ing equations will become 



Ly + M r/ -l=0, L'y + Ny,,-l=0, 



or the centre will move along a right line. 



Again, perpendiculars are let fall from n points on a plane, the 

 sum of the squares of which is constant, the plane will envelope an 

 ellipsoid. 



Should the sums of the squares be varied, the successive surfaces 

 will all be confocal ellipsoids. 



To show that if two surfaces of the second order are enveloped by 

 a cone, they may also be enveloped by a second cone. 



Let the vertex of the cone be taken as the origin of coordinates, 

 and let their tangential equations be 



and as the common tangent planes must pass through the origin, 



