MR. R. LACHLAN - ON SYSTEMS OF CIRCLES AND SPHERES. 
581 
208. If the four spheres meet in a point, the sphere which cuts them orthogonally 
will be coincident with this point, and so the radius must be zero. Hence, if the 
system (1, 2, 3, 4) have a common point, we must have 
( 220 ) 
Five Spheres having a Common Orthogonal Sphere. —§§ 209-211. 
209. Suppose the system of spheres (1, 2, 3, 4, 5) have a common orthogonal sphere, 
x say; then, y denoting any other sphere, the equation 
leads at once to the condition 
/x, 1, 2, 3, 4, 5\ 
n L 1,2,3, 4,5)“ 0 ’ 
n 
1, 2, 3, 4, 5 
1, 2, 3, 4, 5 
; 0 ; 
( 221 ) 
which is the necessary and sufficient condition that the system may have a common 
orthogonal sphere. 
Similarly, if (6, 7, 8, 9, 10) denote any other system of spheres, we should have, 
since 
and hence 
n 
6, 7 , 8, 9, 10\ 
J/, 1 , 2, 3, 4, 5 ) ’ 
n 
/6, 7, 8, 9, 10\ 
Vh 2 , 3, 4. 5 / 
/6, 7, 8, 9. 10\ 
VI, 2. 3, 4, 5 / 
=n 
1,2,3, 4, 5\ /6, 7, 8, 9, 10\ 
1, 2, 3, 4, 5/ X \6, 7, 8, 9, 10/ 
210. It is easy to prove, that if the system of spheres (1, 2, 3, 4, 5) be such that 
the condition (221) is satisfied, then any four of them will be connected with any 
four other spheres (6, 7, 8, 9) by the relation 
/6, 7, 8, 9\ 
\b 2, 3, 4/ 
/l, 2, 3, 4\ 
\b 2, 3, 4j 
xn 
/6, 7, 8, 9\ 
\ 6 , 7 , 8 , 9 )■ 
2 L1 Suppose now (x) to denote any sphere, then the equation 
