THE IN-AHH-CIECHMSCEIBED POLYGON. 
229 
then that the last-mentioned system is equivalent to the transcendental equation 
Ild=Tlp+Wc, 
in which the arbitrary constant which should have been inserted, and the sign of Il^l, are 
determined by the consideration that for k=co (which gives 11^=0) we ought to have 
0=p, and therefore 
There is of course a similar equation in and the terms with Hk must be taken 
with opposite signs, and we have thence the theorem, 
“ If 6, & are the parameters of the points P, P' in which the conic V= 0 is inter- 
sected by the tangent, the parameter whereof is p, of the conic U-|-W=0, then the 
equations 
= Ilp — Wc^ 
116’ =Ilp-\-TUc, 
determine the parameters 6, 6’ of the points in question.” And again, 
“ If the two variable parameters 6’ are connected by the equation 
U6’-Il6=2m, 
then the line PP' will be a tangent of the conic U-j-W=0.” 
The foregoing demonstration relates to the particular forms X^ = ax^-];-h'if-\-cz, 
Y •, but obser\ing that the function which enters 
under the integral sign in the transcendental function HI, is the square root of the dis- 
criminant of U+iV, the theory of covariants shows at once that the conclusions apply 
to any forms whatever of U, V, the expression for the transcendental function being 
V disct. (U + ^V) 
Consider now a triangle inscribed in the conic V=0, and with its sides touching the 
conics 
V+kY =0, 
U-f^V=0, 
U-i-OT = 0. 
Then if 6, 6" are the parameters of the angles, we have 
116” -U6' =2m , 
m’ -m =2nF, 
and thence 
ny?:-fnA:'+nF=0 
as the relation which must subsist between the parameters k, k'y k!\ of the conics touched 
by the sides ; and similarly for a polygon of n sides, the relation between the para - 
meters is 
n^i -|“ 11^2 d” • • • — 0. 
