442 



the type of many of its general forms might be found in earlier 

 examples. 



Mr. Petrie then proposed that the thanks of the Academy 

 should be given to the Rev. Richard Butler, for his kindness 

 in sending this brooch for the inspection of the Meeting. 

 The thanks of the Academy were accordingly voted to Mr. 

 Butler. 



Mr. Ingram read the following note on certain Properties 

 of the Surfaces of the Second Degree. 



" Mr. Salmon, Fellow of Trinity College, has given a 

 mode of generating certain of the surfaces of the second degree, 

 which is in a remarkable way supplementary to the modular 

 method of Professor Mac CuUagh, and which has been called, 

 for distinction's sake, the umbilicar method. In it the surface 

 is had as the locus of a point moving so that the square of its 

 distance from a fixed point is proportional to the rectangle un- 

 der its distances from two fixed planes. Out of this generation 

 arise many highly interesting properties of the surfaces in 

 question, to some of which it is the object of the present com- 

 munication to call the attention of the Academy. 



" The fixed point is called the Focus of the surface, the 

 two fixed planes the Directive Planes, and their line of inter- 

 section the Directrix. 



♦' 1. Two right lines, reciprocal-polars with relation to the 

 surface, meet a directive plane in two points such that the 

 vectors drawn to them from the focus are at right angles. 



" 2. A similar theorem holds for two conjugate tangents at 

 any point of the surface. 



"3. Two right lines, reciprocal-polars with relation to the 

 surface, seen from the pole of a directive plane, appear to cut 

 at right angles. 



" 4. Let a cone be described, passing through two plane 

 sections of the surface ; it will intersect a directive plane in a 

 certain conic : let a second cone be described, passing through 



