«Sii 
PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON, &C. 
15 
[489 
489] PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON, &C. 15 
IBED POLYGON, 
5 ON A CONIC. 
that is 
Pxy — Q (x + y) + R =0, 
or representing this by 
+ Qv + 
we have f : rj : % = xy : —x — y : 1; and consequently (£, rj, £) are connected by a 
quadric equation ; that is, the envelope is a conic. 
The relation (*l[x, l) 2 (y, l) 2 = 0, whether symmetrical or not, leads as will be 
presently shown to a differential equation of the form 
dx , dy A 
vm ± v(F) -0, 
where X, Y are quartic functions of x, y respectively ; viz. these are unlike or like 
functions of the two variables according as the integral equation is not or is sym 
metrical in regard to the two variables. In the former case, however, the functions 
X, Y are so related to each other, that the two can be by a linear transformation 
converted into like functions of the variables : for instance, if y be changed into 
gm/j + b 4- cy-L -1- d, then the constants may be determined in suchwise that Y is the 
cs, vol. xi. (1871), 
same function of y 1 , that X is of x ; the original integral equation being hereby 
converted into a symmetrical equation (*$)», l) 2 {y x , l) 2 = 0 between x and y x , so that 
in one point of view the unsymmetrical case is not really more general than the 
symmetrical one. It is to be added that the integral equation contains really one 
more constant than the differential equation (this is most readily seen in the sym- 
that is perfectly well 
be put together; and 
rrds referred to, some 
metrical case, the differential equation depends only on the ratio of five constants 
a, b, c, d, e, whereas the integral equation depends on the ratio of six constants), so 
that the integral equation is really the complete integral of the differential equation. 
Attending now to the symmetrical case; if A and B are corresponding points, 
then the corresponding points of B are A and a new point C ; those of G are B 
ndation in the theory 
; viz. a (2, 2) corre- 
there correspond two 
spondence either point 
of them will then be 
thing, if x, y are the 
are connected by an 
in regard to the two 
is easy to show that 
ition may be expressed 
bes (P, Q, R) fixed in 
P:Q:R= 1 : x : x 2 , 
le line joining the two 
and a new point D, and so on; so that the points form a series A, B, G, JD,...; 
and the porismatic property is that, if for a given position of A this series closes at 
a certain term, for instance, if D = A, then it will always thus close, whatever be 
the position of A. And this follows at once from the consideration of the differential 
,. dx dy . . . . ., . 
equation ^ ^ = ^ y)’ V1Z ' aS ™ 1S 1S 0nCe m ^ e ^ ra ‘ :, ^ e P er se m f° rm 
n O — II (x) = II (k), 
this equation must be a transformation of the original equation (#$A l) 2 (y, l) 2 = 0, and 
equally with it represent the relation between the parameters x, y of the two points 
A, B; the constant of integration k is of course completely determined in terms of 
the coefficients of the last-mentioned equation, assumed to be given. 
Hence forming the equations for the correspondences, B, C; G, D ;... and assuming 
that the series closes F, A ; we have 
H O — n (y) = n (k), 
II (x) - n (u) = n (k)