541 
of Edinburgh, Session 1881-82. 
ellipse p" , then the sum p + p" = p will represent a surface which 
we may call a hyperbolic Tore , that is a hollow canal having 
everywhere the same elliptical section, parallel namely to the plane 
j , k, of the ellipse p" , and equal to that ellipse; whereas the median 
line of the canal will be the hyperbola p in the plane i, k. 
The two tores corresponding to the two branches of the same 
hyperbola will be symmetrical in respect to the plane i, j , passing 
through the centre (at the origin 0). The two sheets of the same 
surface may intersect or not according as the value of h is outside 
the interval between a 2 and b 2 or inside that interval. 
We will not here undertake to prove that through every point of 
space there must be four surfaces passing, corresponding to four 
different values of h ; we will only remark that the same hyperbola 
will correspond to two different values of h , namely, h and h' satisfy- 
ing to 
h - d 2 = a 2 - h ' . 
To these values correspond two different ellipses. Vice versa, to • 
h-b 2 = b 2 -h", 
for the same ellipse correspond two different hyperbolas . . . 
All the hyperbolas have the same pair of asymptotes; these lines 
pass through the origin and have the directions of S, S', defined by 
a8 = ic + kb , aS’ = - ic + kb . 
Considering now the cases of h = a 2 and h — b 2 , we must re- 
member our assuming of the inequality 
a>b . 
In the case h = a 2 the two tores transform themselves into two right 
cylinders, having their axes coincident with the asymptotes just 
mentioned; their radius perpendicular to the axis = fa 2 -b 2 = c. 
This will easily appear when we consider the value of z becoming 
for h = a 2 
cz= ±bx±ajc 2 -y 2 , 
hence 
(40) (cz =f bx) 2 + ahf- = a 2 c 2 . 
