of Edinburgh , Session 1881-82, 545 
Before writing the equation we remark that the arbitrary 
quantities 
cio , and a J 1 - w 2 , 
when substituted into the place of y and z respectively in the equa- 
tion (41) of the ellipse, satisfy that equation identically. We 
introduce the coordinates y 2 , z 2 , as those of a point of the ellipse, 
putting x 2 — 0 , and 
y 2 = cw , z 2 = a rjl - iv 2 • 
Also for distinguishing the coordinates of the hyperbola from those 
of any point in space we replace x , z! , in (42) by x 1 , z 1 , putting 
?/i = 0. 
The equation in k' now takes the form 
we have 
(45) // = l(<r 2 -o-i) • 
This establishes the second part of Minding’ s Theorem , namely, 
all the single resultants pass through the ellipse because every point 
of the hyperbola is the summit of a cone, of which the generating 
lines are the single resultants passing through that point, and the 
ellipse forms the oblique base of the cone, and vice versa every 
point of the ellipse is the summit of another cone of which the base 
is formed by the two branches of the hyperbola : moreover, all these 
cones are right cones , the axes being the tangent to the curve at the 
point where the summit of the cone is situated, — this latter pro- 
position results from the equation of the cone,, as, for example, in 
