C. II. 
19 
[ 128 1.291 
figure 81 put into 
145 
129. 
ON THE PORISM OF THE IN-AND-CIRCUMSCRIBEU TRIANGLE, 
AND ON AN IRRATIONAL TRANSFORMATION OF TWO TER 
NARY QUADRATIC FORMS EACH INTO ITSELF. 
[From the Philosophical Magazine, voi. ix. (1855), pp. 513—517.] 
quadrilateral ABC I 
pass two and two 
scribed conics, and 
IraAvn, the tangents 
e case of the two 
There is an irrational transformation of two ternary quadratic forms each into 
itself, based upon the solution of the following geometrical problem, 
Given that the line 
lx + my + nz = 0 
meets the conic 
(a, b, c, /, g, K$x, y, zf = 0 
in the point (x x , y x , z x ); to find the other point of intersection. 
The solution is exceedingly simple. Take (x 2 , y 2 , z 2 ) for the coordinates of the 
other point of intersection, we must have identically with respect to x, y, z, 
(a, ...]£#, y, zf.ffl, ...$7, to, nf — k (lx + my + nzf 
= (a,... f[x 1 , z&x, y, z).(a, ...$> 2 , y 2 , z$x, y, z) 
to a constant factor pres. 
Assume successively x, y, z = gt, p^, ®; p^, 23, ; (3i, it follows that 
x 2 : y 2 : z 2 = y x z x {g* (&,... $7, to, nf - (@U + Jfym + <£inf} 
: z x x x {23 (iH,... $7, to, nf — (p^i + 23to + $nf] 
: cc x y x {<& (0,... $7, to, nf — ((&l + JpTO + (&nf\;