ROYAL IRISH ACADEMY. 
Ill 
Lord Talbot de Malahide presented fourteen several volumes of the 
works of Dominic Sestini on Numismatics, printed at Milan, Florence, 
Pisa, and Berlin. 
MONDAY, FEBRUARY 22, 1858. 
James Henthorn Todd, D. D., President, in the Chair. 
The Rev. George Salmon read a paper by Mr. Cayley,— 
on the theory of reciprocal surfaces. 
The present note is intended to be supplementary to Mr. Salmon’s me¬ 
moir “On the Degree of a Surface reciprocal to a given one” (Trans. 
R. I. A., vol. xxi. pp. 461-488; 1857). I find that Mr. Salmon’s equa¬ 
tions admit of a transformation which appears important in reference to 
the geometrical theory, and the object of the note is to present the sys¬ 
tem of equations under the new form. 
Mr. Salmon writes— 
n, the order of the surface. 
a, the order of the tangent cone drawn from any point to the sur¬ 
face. 
£, the number of the double edges of the cone. 
ac, the number of its cuspidal edges. 
b, the order of any double curve upon the surface. 
k, the number of apparent double points of the double curve. 
t } the number of triple points on the double curve. 
c, the order of any cuspidal curve on the surface. 
h, the number of apparent double points of the cuspidal curve. 
/3, the number of intersections of the double and cuspidal curves 
which are stationary points on the cuspidal curve. 
7 , the number of intersections which are stationary points on the 
double curve. 
i, the number of intersections which are not stationary points upon 
either curve. 
P , the number of the points where the double curve is met by the 
curve of contact of the tangent cone. 
<r, the number of the points where the cuspidal curve is met by the 
curve of contact. 
And the accented letters denote the corresponding singularities of the 
reciprocal surface, or, if we choose that they should refer to the given sur¬ 
face, and its tangential or class singularities, then we have— 
n', the class of the surface. 
a r , the class of the curve of intersection by any plane. 
#, the number of double tangents of the curve. 
k, the number of its cusps. 
b', the class of the node-couple develops. 
