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 Cullagh, 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 
