[175 
175] ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 75 
class 4, 
rk that 
If a triangle be inscribed in a conic, and two of the sides touch conics having 
double contact with the circumscribed conic, then will the third side touch a conic 
viz. a 
3 other 
mstruct 
pect to 
.he two 
angents 
wo and 
nd the 
points) 
having double contact with the circumscribed conic. 
Secondly, the conics 21 and 33 may cut the conic © in the same four points. 
Here it may be seen that there are an infinity of inscribed quadrilaterals of the 
kind first considered, viz. of which two opposite sides touch the conic 21, and the 
other two opposite sides touch the conic 33. Hence, the curve (5 is made up of 
two coincident curves of the class 4. But the curve of the class 4 has in fact 4 
double tangents, viz. considering each of the points of intersection of 21, 33, ©, and 
drawing tangents to 21 and 35 meeting © in two new points, the line joining these 
points is a double tangent of the curve in question, which is therefore of the 4th 
33 has 
points 
double 
order, and being of the class 4 with 4 double tangents, it must break up into two 
curves of the second class, i.e. into two conics. Each of these conics passes through 
the points of intersection of 21, 33, ©, and touches the four lines last referred to, 
the conics would of course be completely determined by the condition of passing 
curves 
contact 
which 
5 21 is 
for the 
Dect to 
points, 
double 
icident 
hrough 
conics, 
tion of 
through the four points and touching one of the four lines. Attending only to one 
of the two conics, we have thus what I call the porism (allographic) of the in-and- 
circumscribed triangle, viz. 
If a triangle be inscribed in a conic, and two of the sides touch conics meeting 
the circumscribed conic in the same four points, the remaining side will touch a 
conic meeting the circumscribed conic in the four points. 
The a posteriori demonstration of these theorems will form the subject of an 
other paper, [178]. 
of the 
But 
S have 
m the 
conics, 
igents, 
ler 8; 
curves 
© in 
nts of 
double ] . 
Gt is 
sm of 
n-and- 
10—2