NOTE ON THE CAKTESIAN. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xn. (1873), 
pp. 16—19.] 
The following are doubtless known theorems, but the form of statement, and the 
demonstration of one of them, may be interesting. 
A point P on a Cartesian has three “opposite” points on the curve, viz. if the 
axial foci are A, B, G, then the opposite points are P a , P b , P c where 
P a is intersection of line PA with circle PBC, 
P b „ „ PB „ PCA, 
Pc „ „ PC „ PAB. 
And, moreover, supposing in the three circles respectively, the diameters at right angles 
to PA, PB, PC are act!, ¡3/3', yy respectively, then the points a, a', /3, /3', 7, 7' lie by 
threes in two lines passing through P, viz. one of these, say Pa/3y, is the tangent, 
and the other Pa'/3'7' the normal, at P; and then the tangents and normals at the 
opposite points are P a a and P a A, P b /3 and P 6 /3', P c 7, and P c y' respectively. 
There exists a second Cartesian with the same axial foci A, B, G, and passing 
through the points P, P a , P b , P c (which are obviously opposite points in regard 
thereto); the tangent at P is Pol ¡3'f and the normal is Pa/3y; and the tangent and 
the normal at the other points are P a a and P a a, P b /3' and P b /3, P c y' and P c y respec 
tively: viz. the two curves cut at right angles at each of the four points. 
Starting with the foci A, B, C and the point P, the points P a , P b , P c are con 
structed as above, without the employment of the Cartesian; there are through P 
with the foci A, B, G two and only two Cartesians; and if it is shown that these 
pass through one of the opposite points, say P b , they must, it is clear, pass through