DE. J. CASEY ON CYCLIDES AND SPIIEEO-QITAETICS. 
687 
(compare art. 19). As an example, three lines in space determine a ruled quadric; to 
which we have the corresponding theorem : — Three circles in space orthogonal to the 
same sphere determine a cyclide. Again, every ruled quadric has two systems of recti- 
linear generators ; to this corresponds the theorem : — Every cyclide has two systems of 
circular generators corresponding to each sphere of inversion; or any four rectilinear gene- 
rators of one system on a ruled quadric are cut equianharmonically by all the rectilinear 
generators of the opposite. Hence any four circular generators of one system belonging to 
a cyclide are cut equianharmonically by all the circular generators of the opposite system. 
265. From recent articles we see that, being given any graphic property of the focal 
quadric F of a cyclide W, we can get a corresponding property of W by the following 
substitutions : — 
l.< * 
i 
cl 
l 
'a. 
b. 
c. 
cl 
F 
W 
For a point on F, pi 
A point having any special 
relation to F, 
A system of complanar points, -< c. 
A system of collinear points, 
cl 
A tangent line to 
A line having any special rela- 
tion to 
A system of concurrent lines, 
A system of complanar lines, 
f a. 
b. 
d. 
A generating sphere of W. 
A sphere having a corresponding rela- 
tion to W. 
A system of spheres through the same 
two points. 
A system of spheres through the same 
circle. 
A circle having double contact with 
A circle having a corresponding relation 
to 
A system of homospheric circles. 
A system of circles through the same 
two points. 
f a - 
b. 
III. < c. 
cl 
l 
A tangent plane to 
A plane having any special 
relation to 
A system of planes through the 
same point, 
A system of planes through the 
same line, 
r«- 
I bm 
\ c . 
cl 
A pair of inverse points on 
A pair of inverse points having a corre- 
sponding relation to 
A system of homospheric pairs of inverse 
points. 
A system of inverse pairs of points on 
the same circle. 
266. In order to give illustrations of this system of substitutions, I give the theorems 
derived by them from a splendid paper of Mr. Townsend’s in the ‘ Quarterly Journal,’ 
vol. viii. p. 10. For this purpose the following proposition is necessary: — If three 
tangent planes to a quadric be mutually perpendicular the locus of their point of inter- 
section is a sphere, called the director sphere of the quadric. Hence, by substitution, we 
get the following theorem : — If three lines mutually 'perpendicular be drawn through a 
centre of inversion of a cyclide , and if P, P', Q, Q', E, E be three pairs of inverse points 
