DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
C03 
of the curve WO on the planes through the intersection of O and a, O and /3, &c. be 
denoted by x, y , 2 , w ; then if the distances from the centre of O to the centres of a, /3, y, § 
be denoted by <y 2 , a> 3 , respectively, it is easy to see that the results of substituting 
the coordinates of P in a, f3, y, 0 are 2oj 1 x, 2 a. 2 y, 2^ 3 z, 2 &> 4 w, and therefore the quadric 
au\x"~ -f- boff + cu\zr -j- daw— 0 (39) 
passes through the curve WO. Hence the curve WO is also the intersection of the 
sphere O and the quadric (39), and therefore it is a sphero-quartic. 
38. If in the last article we suppose the sphere O to coincide with U, the sphere 
orthogonal to a, /3, y, c$, and if we denote the circles in which the spheres «, /3, y, 
intersect Uj by the same notation, that is, by a, (3, y, (5, then if the radii of the circles 
a, (3 , y, be /, /', /", /'", it is plain that <y 2 , a> 3 , co i of the last article become sec/, 
sec /', sec r"’, sec /'", the radius of U being denoted by unity. Hence, by articles 36 
and 37, we have the following theorem : — If W=(«, b, c , d , l , m, n , p, q, rfa. (3, y, c5) 2 =0 
be the equation of a cyclide , the equation of the sphero-quartic WU will be 
( a , b, c , d, l, m, n, p, q, r) 
cos /’ cos /'’ cos /"’ cos r 
Jin 
=o, 
. (40) 
where a , /3, y, 0 are the small circles of intersection of the spheres a , (3, y, 0 with U. 
39. From the three last articles, combined with art. 28, we have at once the following 
theorem, which is a very important one in the theory of sphero-quartics : — If a , (3 , y, 
be any four small circles on the sphere U, the following relation will be true for any point 
on the surface of U, and will therefore be an identical relation : 
-1, 
cos («/3), 
cos (ay), 
cos (a^), 
a A- sin /, 
cos (/ 3u ), 
-1, 
cos ((3 y), 
cos (/3t$), 
/3 a- sin /', 
cos (y«) , 
cos (y/3), 
-1, 
cos (y!$), 
y A- sin /", 
cos (ba) , 
cos (b(3 ) , 
cos (^y) , 
-1, 
ci A- sin/'", 
a A- sin/, 
/3 a- sin/', 
* III 
y -f- Sin T , 
0 a- sin /"', 
0 , 
Cor. If the circles a , f3, y, § on the surface of U be mutually orthogonal, the relation 
is identically true for any point on U, 
1 y* 
sin 2 r 1 ‘ siu 2 r" ‘ sin 2 r " 1 
snr r 
8 i -0 
2 Jill U • 
• • ( 42 ) 
If we incorporate the constants with the variables this becomes « 2 +/3 2 -f-y 2 -|-^ 3 =0. 
40. If a, (3, y be three small circles on the sphere U, and if a sphero-quartic be given 
by an equation of the second degree ( a , b, c,f g, hfa, (3, y) 2 = 0, I say the tangential 
equation of the corresponding focal sphero-conic is 
( a , b , c,f g, hfh, [x, f — 0 (43) 
4 0 
MDCCCLXXT. 
