[114 
115] 
87 
t /i 
№ ® 
\ 33 JF 
i JF ® 
should be some 
in the previous 
115. 
NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED 
POLYGON. 
[From the Philosophical Magazine, vol. VI. (1853), pp. 99—102.] 
The equation of a conic passing through the points of intersection of the conics 
is of the form 
U= 0, F=0 
wU + V = 0, 
where w is an arbitrary parameter. Suppose that the conic touches a given line, we 
have for the determination of w a quadratic equation, the roots of which may be 
considered as parameters for determining the line in question. Let one of the values 
of w be considered as equal to a given constant k, the line is always a tangent to the 
conic 
kU+F= 0; 
and taking w—p for the other value of w, p is a parameter determining the particular 
tangent, or, what is the same thing, the point of contact of this tangent. 
Suppose the tangent meets the conic U= 0 (which is of course the conic corre 
sponding to w = oo) in the points P, P', and let 6, oo be the parameters of the point 
P, and 6', oo the parameters of the point P'. It follows from my “Note on the 
Geometrical representation of the Integral jdx -f- *J(x + a){x-\- b) (x + c), [113] ( J ) and 
from the theory of invariants, that if □ £ represent the “Discriminant of IjU + F 
1 I take the opportunity of correcting an obvious error in the note in question, viz. a- + b i + c?-2bc-2ca-2ab 
is throughout written instead of (what the expression should be) b 2 c 2 + c 2 a~ + a 2 b--2a i bc-2b' 2 ca-2c 2 ab. [This 
correction is made, ante p. 55.]