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XXI. On Cy elides and Sphero- Quartics. 
By John Casey, LL.JD., M.R.I.A. Communicated by A. Cayley, F.B.S. 
Received May 11, — Read June 15, 1871. 
CHAPTER I. 
Section I. — Spheres cutting orthogonally . 
Art. 1. If x ^ Ay ^ AAA ^ yxA ^ f y - p ^ hz-\-c =0, 
cc 2 Ay 1 -f- 3 2 A 2 < 7 ^-j- 2 f'yA 27^2 + d = 0 
be the equations of two spheres, these spheres will intersect orthogonally if the square 
of the distance between their centres be equal to the sum of the squares of their radii. 
Hence we infer that the two spheres whose equations are given above will intersect 
orthogonally if the condition holds, 
+ ......... ( 1 ) 
2. From art. 1 we can easily find the equation of a sphere cutting orthogonally four 
given spheres, S', S", S'", S"". Thus, if the given spheres be 
x 1 A l/ 2 + 2 2 + 2 g'xA 2^2 +c=0 &c. , 
the equation of the orthogonal sphere is 
x 2 Ay 2 A“ 2 , 
0C 5 
y . 
z , 
1 
d , 
-/' . ■ 
-K , 
1 
c" , 
-9\ 
-/"•> ■ 
-h" , 
L 
d", • 
-f. 
-1C' , 
1 
c"", . 
-1C", 
1 
( 2 ) 
Cor. If a sphere S cuts four spheres, S', S", S'", S"", orthogonally, it also cuts ortho- 
gonally AS'+|W(S" + j'S"'-j-g i S"" when A, g-, v, § are any multiples. 
3. The following method finds the equation of the orthogonal sphere in tetrahedral 
coordinates. Let S', S", S'", S"" be the given spheres, then AS' -j- (JjS" -f- vS"' + ^S"" is coor- 
thogonal with S', S", &c. ; and if the radius of AS'+^S" + {'S"'-j-gS"" be evanescent, its 
centre must be a point on the required orthogonal sphere ; but if its radius be zero, it 
represents an imaginary cone and the discriminant vanishes. It is easy to see that 
A, yj, v , § are the tetrahedral coordinates of the centre of AS' + ^S"+yS'"-j-§S ; '", the tetra- 
hedron of reference having its angular points at the centres of S', S", &c. 
Now let the spheres S', S", &c. be given in the form 
(x — a’) 2 A(y— V) 2 + ( z — c') 2 — r 2 = 0 See., 
