THE IN-AND-CIECUMSCEIBED POLTaON. 
227 
whereof are such that (?=0 is the relation for the triangle, «Z=0 the relation for the 
quadrangle; thus [3]=c, [4] = <Z, and for the particular cases considered, the analytical 
theorem consists herein, that c is a factor of (6), and of (9), and that is a factor of (8). 
I have, for the sake of homogeneity, introduced into the formulae the quantity a { = 1), 
but this is a matter of form only. 
The functions [3], [4], &c. have been spoken of ?kS, 'prime they are so, in fact, as far 
as they are calculated ; and that they are so in general rests on the assumption that for 
a polygon of a given number of sides, there is but one form of relation : if, for instance, 
in the equation [12] = 0, which is the condition for a proper dodecagon, the function 
[12] could be decomposed into rational factors ; then equating each of these factors to 
zero, we should have so many distinct forms of relation for a proper dodecagon. I 
beheve that the assumption and reasoning are valid ; but without entering further into 
this, I take it for granted that in the general case the functions [3], [4], &c. are in fact 
prime. But the coefficients j3, y, or h, c, d, instead of being so many independent 
arbitrary quantities, may be given as rational functions of other quantities (if, for 
instance, the two conics are circles, radii R, r, and distance between the centres a, then 
/3, 7 , S will be functions of R, r, a) : and it is in a case of this kind quite conceivable that 
the functions [3], [4], &c., considered as functions of these new elements, should cease 
to be prime functions. In fact, in the case just referred to of the two circles (the 
original case of the Porism as considered by Fuss), the functions [4], [6], &c., which 
correspond to a polygon of an even number of sides, appear to be each of them decom- 
posable into two factors : the memoir contains some remarks tending to show d priori 
that in the case in question this decomposition takes place. I was led to examine the 
point by the elegant formulae obtained in an essentially different manner by M. Men- 
tion, Bull, de I’Acad. de St. Pet. t. i. pp. 15, 30, and 507 (1860), in reference to the 
case of the two circles (it thereby appears that the decomposition takes place for the 
quadrangle and the hexagon) ; and these formulae are reproduced in the memoir. 
I. 
Demonstration of the general Formula of the Delation between the two Conics. 
The equation of a conic passing through the points of intersection of the conics 
U=0, V=0, 
is of the form 
U+mV=0, 
where m is an arbitrary parameter. Suppose that the conic touches a given line, w^e 
have for the determination of m a quadratic equation ; and conversely, if the roots of 
this quadratic equation are given, the line is also given ; that is, the two roots may be 
considered as parameters which determine the particular line. 
Let ^ be a given value of m ; the parameters of any tangent of the conic U -1-A:V = 0 
are p, but as 1c is common to all the tangents, we may consider the particular tangent 
2 I 2 
