554 
[380 
380. 
NOTE ON THE RECT AN GULAK HYPERBOLA. 
[From the Oxford, Cambridge and Dublin Messenger of Mathematics, vol. I. (1862), p. 77.] 
Every conic which passes through the points of intersection of two rectangular 
hyperbolas is a rectangular hyperbola. In fact if a conic be referred to rectangular 
axes, the condition that it may be a rectangular hyperbola is Coeff. of x- = — Coeff. of y-. 
Hence if U, V be any two quadratic functions of x, y, and if X be a constant, the 
condition in question being satisfied for each of the functions U, V, is satisfied for 
the function U+ \V: and the equation of any conic through the points of intersection 
of the conics U= 0, V=0 is C r + \I r =0: which proves the theorem in question. 
In particular if from two of the angles of a triangle perpendiculars are let fall 
on the opposite sides, and if the point of intersection of the perpendiculars and the 
third angle be joined: then since the first side and the perpendicular upon it are a 
rectangular hyperbola, and the second side and the perpendicular upon it are a 
rectangular hyperbola; the third side and the joining line must be a rectangular 
hyperbola: that is, these two lines must be at right angles to each other. We have 
thus the well-known theorem that the perpendiculars let fall from the angles of a 
triangle on the opposite sides meet in a point. 
The theorem as to the hyperbolas is a particular case of the theorem that three 
conics which pass through the same four points are met by any line whatever in 
six points forming a system in involution. In fact a rectangular hyperbola is a conic 
meeting the line at infinity in two points harmonically related to the circular points 
at infinity: hence two of the conics being rectangular hyperbolas, the foci of the 
involution are the circular points at infinity: hence these points and the points in 
which the line at infinity meets the third conic are harmonically related to each other; 
that is, the third conic is a rectangular hyperbola.