C. III. 
7 
49 
171 J NOTE ON MR SALMON’S EQUATION OE THE ORTHOTOMIC CIRCLE. 
V = A V+ BH(V), i. e. A = 1, P = 0: the latter equation determines, what is alone 
important, the ratio b : a; the equation is of the third order, so that there are in 
general three distinct solutions TJ = a V + BH ( V) = 0. 
In the particular case in which the curves of the third order V= 0 is made up 
of a line P = 0 and a conic W = 0, i. e. where V = P W = 0, the curve H (V) = 0 is 
made up of the same line P= 0 and of a conic having double contact with the conic 
W = 0 at the point of intersection with the line P = 0, i.e. H (P W) = P (IW + mP 2 ). 
And U = aPW-hbH(PW) is consequently a function of the same form, i.e. the cubic 
TJ = 0 is made up of the line P = 0 and of a conic having double contact with the 
conic W = 0 at its points of intersection with the line P — 0. We may therefore write 
TJ = P {/W + gP 2 ), and forming with this value the equation PW = P (FW + GP 2 ), it 
may. be noticed that, owing to the occurrence of a special factor which may be rejected, 
the resulting equation G — 0 gives only a single value for the ratio f : g. Forming 
from the value TJ = P (fW + gP 2 ), the equation A, ~ + g + v — 0, the equation 
thus obtained will be of the form W + PQ = 0, which is the equation of a conic 
passing through the points of intersection of the line and conic P = 0, W = 0, and 
besides intersecting the conic W = 0 in two other points. And it may also be shown 
that the four points of intersection, (i. e. the points given by the equations W = 0, 
W J rPQ = 0), the pole of the line P — 0 with respect to the conic W = 0, and the pole 
of this same line with respect to the conic W + PQ = 0, lie all six in the same conic. 
We see, therefore, that, given a curve of the third order, the aggregate of a line 
P — 0 and a conic W = 0, as the locus of the point such that its polars, with respect 
to three several conics (or a system depending on three conics), meet in a point, each 
conic of the system is a conic passing through the points of intersection of the line 
and conic P = 0, W — 0, and, moreover, such that the four points of intersection with 
the conic W = 0 and the poles of the line P = 0, with respect to the conic of the 
system, and with respect to the conic W = 0, lie all six in the same conic. In the 
particular case in which the line and conic P = 0, W = 0 are the line at oo and a 
circle, each conic of the system is a circle such that its points of intersection with 
the circle W = 0 and the centres of the two circles lie in a circle, i. e. the conics are 
circles cutting at right angles the circle W — 0, which agrees with Mr Salmon’s theorem. 
To verify the assumed theorems in the case of the curve of the third order 
V = P W = 0, we may take 
P = ax + /3y + <yz = 0 
for the equation of the line, and 
W = x 2 + y 2 + z 2 = 0 
for the equation of the conic. I write, for greater convenience, U = P (£ fW + 7 gP 2 ) ; 
the Hessian of this is 
f(P +2ax) + ga 2 P, 
/(#»+ ay) + ga(3P, 
f{az + yx) + gya P, 
/(#»+ a y) + f J a PP, 
f(P +2/3y)+g/3 2 P, 
f(yy+ fiz) + gfiyP, 
f(az +yx)+gya P 
f (yy + f3z ) + gfiyP 
f{P + 27^) + gy 2 P