56 
NOTE ON THE GEOMETRICAL REPRESENTATION &C. [113 
i. e. if the points P, P' upon the conic x 2 + y 2 + z 2 = 0 are such that their parameters 
6, 6' satisfy this equation, the line PP' will be constantly a tangent to the conic 
k (x 2 + y 2 + z 2 ) + (ax 2 + by 2 + cz 2 ) — 0. 
Hence also, if the parameters k, k\ k" of the conics 
satisfy the equation 
k (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0, 
k' (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0, 
k" (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0, 
n& + tip + nr = 0, 
there are an infinity of triangles inscribed in the conic x 2 + y 2 + z 2 = 0, and the sides 
of which touch the last-mentioned three conics respectively. 
Suppose 2Uk = Hk (an equation the algebraic form of which has already been 
discussed), then 
ne’ -ue = it k, 
6 = oo gives 6' =«; or, observing that 6 = oo corresponds to a point of intersection 
of the conics x 2 + y 2 + z 2 = 0, ax 2 + by 2 -f cz 2 = 0, k is the parameter of the point in 
which a tangent to the conic k (x 2 + y 2 4- z 2 ) + ax 2 + by 2 + cz 2 = 0 at any one of its 
intersections with the conic x 2 +y 2 + z 2 = 0 meets the last-mentioned conic. Moreover, 
the algebraical relation between 6, 6' and k (where, as before remarked, k is a given 
function of k) is given by a preceding formula, and is simpler than that between 
6, 6' and k. 
The preceding investigations were, it is hardly necessary to remark, suggested by 
a well-known memoir of the late illustrious Jacobi, and contain, I think, the extension 
which he remarks it would be interesting to make of the principles in such memoir 
to a system of two conics. I propose reverting to the subject in a memoir to be 
entitled “ Researches on the Porism of the in- and circumscribed triangle.” [This was, I 
think, never written.]