16 



of the operation of taking the'scalar part of a quaternion. The 

 equations (6) shew that v is the vector, of which the recipro- 

 cal v"' represents in length and in direction the perpendicular 

 let fall from the common origin of the variable vectors here 

 considered on the plane which touches at the extremity of the 

 vector p the locus of that variable extremity ; so that v " ' is 

 here a symbol for the perpendicular let fall from the centre of 

 the ellipsoid on the tangent plane to that surface : and v itself 

 denotes, in length and in direction, the reciprocal of that per- 

 pendicular, so that it may be called the vector of proximity of 

 the tangent plane, or of the element of the surface of the ellip- 

 soid, to the centre regarded as an origin. Accordingly, the 

 equation here marked (5) was given in the Abstract of July, 

 1846 (where it was numbered 45), as a formula for determin- 

 ing what was there also called the vector of proximity of the 

 tangent plane of the ellipsoid. It may now be seen that 

 the symbolical connexion between the two equations above 

 marked (6), and the two other equations lately numbered (7), 

 corresponds to, and expresses, in this Calculus, under what may 

 be regarded as a strikingly simple form, the known connexion 

 of reciprocity between any two surfaces, of which one is the 

 loous of the extremities of straight lines drawn from any fixed 

 point, so as to be in their directions perpendicular to the tan- 

 gent planes of the other surface, and in their lengths inversely 

 proportional to those perpendiculars : from the perception of 

 which general relation of reciprocity between surfaces, exem- 

 plified previously for the case of two reciprocal ellipsoids by 

 that great geometrical genius (Professor Mac Cullagh), whose 

 recent and untimely loss we all so deeply deplore, the author 

 of the present communication was led to announce to the 

 Academy, in October, 1832, the existence of certain circles of 

 contact on Fresnel's wave, which he saw to be a necessary con- 

 sequence of the existence of certain conical cusps on another 

 and reciprocal surface. A very elegant geometrical proof of 

 the same general theorern of reciprocity was given afterwards, 



