586 
DK. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
and then the required discriminant will be, after dividing by the factor 
^ ^ -f- ^) { + ^' 2 + ^ ,2 — ^ ,2 ) -l- ^ (a" 3 4- + c" 2 — r" 2 ) 
+ v(a!" 2 + V 112 + c'" 2 - r'" 2 ) + z(ct"" 2 + b"" 2 + c"" 2 - r"" 2 ) \ 
= (a’ X + a"p + d"v + a""?) 2 + (b'k + b"p + VS + b nt Sj 
+ (c'x + c"[M + c'\ + c""tf; 
and this is readily found to be equivalent to the following equation, in which (S' S") &c. 
denotes the angle of intersection of the spheres S', S", &c. : — 
(x/) 2 +(^r") 2 +K') 2 +( ? r' w ) 2 
—2 Xfjjr'r" cos(S'S")— 2 Xvr'r"' cos (S'S"')— 2 Xgr'r"" cos (S'S"") 
-2 prV" cos (S"S" l )—2vgr"'r 1 " 1 cos (S'"S"")-2 ^r""r" cos (S""S") = 0. J 
See my paper “ On the Equations of Circles &c.,” in the Proceedings of the Royal Irish 
Academy, vol. ix. pt. iv. p. 410. 
This equation is simplified by incorporating the radii r\ r", &c. with the variables 
X, [jj, v, g> ; thus put Xr'=x, [Ar"=y, &c., and we get the equation of the sphere orthogonal 
to four given spheres in the form 
x 2J cy 2 _|_ rfj [ _ w 2__2xy cos (S'S") -2 xz cos (S'S'")-2a?w cos (S'S"") 
-2 yz cos (S"S'")-2 zw cos (S'"S"")-2wy cos (S""S")=0. 
Cor. 1. Hence, if the four given spheres be mutually orthogonal, the equation of their 
orthogonal sphere in tetrahedral coordinates is 
(Xr J ) 2 +(^ j y+( V r^+( s r "") 2 = 0 or x 2 +y 2 +z 2 +w 2 = 0 (5) 
Cor. 2. The sphere orthogonal to four given spheres is inscribed in each of the eight 
quadrics, 
U 2 = (Xr'± [ jur"± V 7 J "± §r"") 2 , (6) 
where U denotes the orthogonal sphere f. 
* [The vanishing of the factor ( A + p + v + p ) 2 is the condition that the sphere AS + pS' + v S" + p S'" may become 
a plane. Hence A+p+v+p=0 may he regarded as the tangential equation of the centre of the sphere which 
cuts orthogonally the four spheres S, S', S", S'".' — January 1872.] 
f [Professor Cayley remarks as follows on this article : — “ You give in passing what appears to me an inter- 
esting theorem, when you say ‘ it is easy to see that A, p, v, p are the tetrahedral coordinates of the centre of the 
sphere AS+pS' + vS" + pS"'=0.’ Take any four quadric surfaces S=0, S'=0, S"=0, S"'—0; and establish the 
relation AS+pS' + vS" + pS , "= a C one. This establishes between A, p, v, p and x, y, z, w four linco-linear equa- 
tions, so that, eliminating either set of variables, we have between the other set a quartic equation ; moreover, 
the variables of each set are proportional to cubic functions of the other set (see my “Memoir on Quartic Sur- 
faces,” vol. iii. pp. 19-69 of the Proceedings of the London Mathematical Society). Then your theorem is, that 
when the four quadrics have a common conic, the x, y,z,w and A, p., v, p are linear functions each of the other, 
so that the two quartic surfaces are homographically related, or, by a proper interpretation of the coordinates, 
may be regarded as being one and the same surface.” My theorem, that A, p, v, p are the tetrahedral coordi- 
nates of the centre of the sphere AS-j-pS' + vS" + pS"':=0, is easily proved as follows : the centre of the sphere 
AS + pS' + yS" + pS'" is evidently the mean centre of the centres of S, S', S", S'" for the system of multiples 
A, p,, v, p ; in other words, it is the centre of gravity of four masses proportional to A, p, v, p placed at those 
points. Hence the proposition follows at once by a well-known theorem in Statics. — January 1872.] 
