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NOTICES OF COMMUNICATIONS TO THE 
When the curves 21 and 23 are conics, the curve (5 is therefore of the class 8, 
with 16 double tangents but no inflexions, consequently of the order 24. But there 
are two remarkable cases in which the order is further diminished. 
First Avhen each of the conics 21, 23 has double contact with the conic @. The four 
points of contact give rise to 8 new double tangents or there are in all 24 double 
tangents, the curve (S is therefore of the degree 8: and being of the class 8 with 24 
double tangents, it must of necessity break up into 4 curves each of the class 2, i.e. 
into 4 conics. Each of these has double contact with the conic ©, or attending to only 
one of the four conics we have the well-known theorem which I call the porism 
(homographic) of the in-and-circumscribed triangle, viz. “ there are an infinity of 
triangles inscribed each in a conic, and such that the sides touch conics having each 
of them double contact with the circumscribed conic.” 
Secondly, the conics 21 and 23 may intersect the conic © in the same four 
points. Here every tangent of the curve (£ is in fact a double tangent belonging to 
the first-mentioned group, the curve (S in fact consists of two coincident curves: each 
of them is therefore of the class 4. But this curve of the class 4 has itself four 
double tangents arising from the common points of intersection of the conics 21, 23 
with the conic ©; it must therefore break up into two curves each of the class 2, 
i.e. into two conics: each of these intersects the conic © in the same four points 
in which it is intersected by the conics 21, 23. Attending only to one of the two 
conics we have the other well-known theorem which I call the porism (allographic) of 
the in-and-circumscribed triangle, viz. “ there exist an infinity of triangles inscribed in 
a conic, and such that the sides touch conics, each of them meeting the circumscribed 
conic in the same four points.” 
3. On the Notion of Distance in Analytical Geometry. Report, 1858, p. 3. 
The author remarks that the principles of Modern Geometry show that any 
metrical property whatever is really based upon a purely descriptive property, and that 
these principles contain in fact a theory of distance—but that such theory has not 
been disengaged from its applications and stated in a distinct and explicit form. The 
paper contains an account of the theory in question, viz. it is shown that in any 
system of geometry of two dimensions the notion of distance can be arrived at from 
descriptive principles by means of a conic called the Absolute, and which in ordinary 
geometry degenerates into a pair of points. 
4. On Curves of the Fourth Order haviny Three Double Points. Report, 1860, p. 4. 
The paper is a short notice only of researches which the Author is engaged in 
with reference to curves of the fourth order having three double points. A curve of 
the kind in question is derived from a conic by the well-known transformation of 
substituting for the original trilinear coordinates their reciprocals: and the species of 
the curve of the fourth order depends on the position of the conic with respect to 
the fundamental triangle.