

1862.] 249 



the expression for the volume of the least or central pedal. Amongst 

 the new properties of such pedals the following may he here 

 cited : 



The volume of the pedal whose origin is at a corner of the rect- 

 angular parallelepiped described about the primitive ellipsoid is 

 equal to four times the volume of the central pedal t and to twice the 

 volume of the pedal at any one of the eight points where the ellip- 

 soid is pierced by the diagonals of the parallelopiped. 



Again, the algebraical sum of the volumes of the three ellipsoid- 

 pedals whose origins are at the extremities of any three conjugate 

 diameters of a concentric and co-axal quadric is constant, and equal 

 to three times the volume of the pedal at any one of the eight points 

 where this quadric is pierced by the diagonals of its circumscribed 

 rectangular parallelopiped. 



From this theorem several others are deduced by assuming, for 

 the quadric in question, particular forms. For instance, when it 

 coincides with the primitive surface itself, we learn that the sum of 

 the volumes of the three ellipsoid-pedals whose origins are at the 

 extremities of any three conjugate diameters of the primitive sur- 

 face is constant, and equal to six times the volume of the central or 

 least pedal. 



In this theorem is included, of course, the special case where the 

 origins of the three pedals coincide with the vertices of the primitive 

 ellipsoid. 



If, for convenience of enunciation, we define the pedal-altitude at 

 any point to he the altitude of a parallelopiped whose base is the 

 square on the line joining that point to the centre of the ellipsoid, 

 and whose volume is equal to that of the pedal having the point in 

 question for origin, it is found that the algebraical sum of the three 

 pedal-altitudes at the extremities of any three orthogonal diameters 

 of a quadric, concentric and co-axal with the primitive ellipsoid, is 

 constant, and equal to three times the pedal-altitude at any one 

 of the eight points on this quadric which are equidistant from 

 its axes. It follows, consequently, that this sum is not only inva- 

 riable for one and the same quadric, but for all concentric and 

 co-axal quadrics which pass through one and the same point equi- 

 distant from the principal diametral planes of the primitive ellip- 

 soid. 



VOL. XII. T 



