DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
609 
at the centre of the sphere of inversion U of the cy elide, according as we regard the vari- 
ables a , (3, 7 , § as spheres , as circles on the sphere U, the Jacobian of the spheres a, (3, y, b, 
or as single sheets of cones having their common vertex at the centre of U. 
45. From the double interpretation of the equation +Li sec £=0 as denoting a 
small circle on the sphere, or as denoting a single sheet of the cone (46), all the results 
which we shall prove in the following articles are twofold in their application ; for sim- 
plicity, however, I shall consider it as denoting a circle unless the contrary is expressed. 
If the equation of the plane Lbe ax-\-by-\-cz— 0, it is clear that a, b, c maybe regarded 
either as the direction cosines of L, or the coordinates of its pole on the sphere U, for 
Again, being given 
¥= 
-+7V 
r 
+ w 
r , ( a 2 f Y 
I = I /y? — •' I 
— 1 = 0 , 
- J - 0 ’ 
the equation of the cyclide is 
4 {(« 2 fii)x 2 +(?»“ + /j-pr + ( c - + -j 
2 erf , 2 ¥g , M , „ \ • ........ (B) 
=x 2 +if+z 2 + % J - os+— f .y+— — z+r 2 . 
a Hi +^i C J 
The origin in equation (A) is the centre of J, and the origin in equation (B) is the centre of J', that is, the 
point whose coordinates with respect to the centre of J are 
—Hi f zM, —hJ 
a ~-\~Hi + C “ + Pi 
In order to compare the equations (A) and (B), which represent the same surface, we must transform (B) to 
the same origin as (A), or (A) to the same origin as (B) : we will adopt the latter transformation, and we get 
the following result : — 
4 
= 1 lx + ~, 
Hit V 
+nJ 
(C) 
Since the equations (B) and (C) represent the same cyclide and are referred to the same origin, by comparing 
the absolute terms, we shall get the value of r 3 in terms of r' 2 , y„, and known constants. The absolute term in 
equation (C) is 
a Y ?/ 2 , & 2 
(«> i) s j 
which, being reduced by means of the relation 
f . r 
VJg 2 , c 2 /ifA 2 1 
,3 +Pi) 2 (° 2 +/ J i ) "J ’ 
¥ -=i+4 
+ Hi h 2 -\-g 1 c 2 + /q Hi 
JJi 2 ~r- 
Af + hM 
+ : 
( a2 +HiY (^ 2 +Pi) 2 (c 2 + Hi ) 2 
becomes 
