ON THE POEISM OF THE IN-AND-CIRCUMSCBIBED POLYGON, 
AND THE (2, 2) CORRESPONDENCE OF POINTS ON A CONIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XI. (1871), 
pp. 83—91.] 
The present paper includes, as will at once be seen, much that is perfectly well 
known; but the separate theories required, it seemed to me, to be put together; and 
there are, particularly as regards the unsymmetrical case afterwards referred to, some 
results which I believe to be new. 
The porism of the in-and-circumscribed polygon has its foundation in the theory 
of the symmetrical (2, 2) correspondence of points on a conic; viz. a (2, 2) corre 
spondence is such that to any given position of either point there correspond two 
positions of the other point; and in a symmetrical (2, 2) correspondence either point 
indifferently may be considered as the first point and the other of them will then be 
the second point of the correspondence. Or, what is the same thing, if x, y are the 
parameters which serve to determine the two points, then x, y are connected by an 
equation of the form QPfoc, l) 2 (y, 1) 2 =0, which is symmetrical in regard to the two 
parameters (x, y). In the case of such symmetrical relation it is easy to show that 
the line joining the two points envelopes a conic. For the relation may be expressed 
in the form (*$1, x + y, xy) 2 = 0; we may imagine the coordinates (P, Q, R) fixed in 
such manner that for the point {x) on the first conic we have P:Q:R= 1 : x : x 2 , 
and for the point (y), P : Q : R = 1 : y : y 2 ; the equation of the line joining the two 
points is then 
Q,