248 [Recess, 



to any point on the primitive surface. The primitive surface re- 

 maining unaltered, the form and magnitude of its pedal vary, of 

 course, with the position of the pedal origin. 



In the first part of the memoir, of which the present note is an 

 abstract, the volumes of pedals derived from the same primitive sur- 

 face, but corresponding to different origins, are investigated, arid the 

 general formula found by means of which the volume of any pedal 

 whatever may be calculated when that of any other is known. From 

 this formula are deduced the following new and very general proper- 

 ties of pedal surfaces : 



Whatever may be the nature of the primitive surface, the origins 

 of pedals of the same volume lie on a surface of the third order. 



It should be observed that the volume of the pedal is here under- 

 stood to be that of the conical space swept by the perpendicular, as 

 the tangent plane of the primitive takes all possible positions. In 

 this sense the term volume may clearly be applied to the pedals of 

 unclosed surfaces. It is in fact to such surfaces that the above 

 theorem applies ; for when the primitive is a closed surface, but in 

 other respects perfectly arbitrary, the locus of the origins of pedals 

 of constant volume is a quadric, or surface of the second order. 

 The whole series of quadric loci, corresponding to all possible vo- 

 lumes, constitutes a system of similar, similarly placed, and con- 

 centric quadrics, the common centre of all being the origin of the 

 pedal of least volume. 



From the three equations which determine the position of the ori- 

 gin of the pedal of least volume, it follows that this origin always 

 coincides with the centre of the primitive, whenever the latter pos- 

 sesses such a point ; when, moreover, the primitive, besides being 

 closed, is everywhere convex in curvature, and symmetrical with 

 respect to three rectangular planes, each origin-locus is an ellipsoid 

 whose principal diametral planes coincide with the planes of sym- 

 metry. 



This is the case with the pedals of the ellipsoid, which, ever since 

 the researches of Fresnel on light, have been regarded with especial 

 interest. Their properties form the subject of the second part of 

 the memoir. 



It is shown that the volume of any ellipsoid-pedal, the coordinates 

 of whose origin are given, may be found by simple differentiation of 



