MR. R. LACHLAN ON SYSTEMS OP CIRCLES AND SPHERES. 
587 
o' 
t 3 
h,2 > 
t 3 
h,3 j 
t 3 
h,-± 5 
f 3 
fc 3,l ’ 
0, 
t 3 
l 5 3 
t 3 
^3, 1 > 
hj, 
o, 
hj, 
hJ 
hJ, 
t 3 
H ,3 > 
o, 
t 3 
L,5 
hl\ 
hJ, 
t 3 
6 5,3 5 
0 
= 0 , 
(236) 
which must be satisfied if the spheres all touch the same sphere. Supposing this 
condition satisfied, the radius of the tangent sphere is easily found thus. Let 
(6, 7, 8, 9, 10) denote the system of spheres orthogonal to the spheres (1, 2, 3, 4, 5), 
then by equation (228) we have 
hence 
7r »'.l | rr x,1 | 7r -r,3 | TD-.-t | 77 X 5 Y 
^l.G 7r 3,8 7r 4,9 ’L.IO 
+ 
Ji + —■+—+ 
7r 2,7 ^3,8 ’Ft.O 
(237) 
where the radii r 2 , &c. are to be taken with the positive or negative sign, according 
as the contact is external or internal. 
Chapter III.— Spheres Connected with a System oe Four Spheres. 
In this chapter it is proposed to extend a few of the results arrived at in 
Chapter IV., Part I. The formulae for the spheres which pass through the points of 
intersection of four spheres, and also for the spheres which touch four spheres, will be 
seen to be exactly analogous to those proved for circles passing through the points of 
intersection of, and touching three given circles. In the case of a tetrahedron there 
does not seem to be any sphere analogous to the nine-points circle of a triangle; a 
special system of spheres will, however, be mentioned, which have a sphere touching 
their tangent spheres, but even here there is no formula, connecting the radius of this 
sphere with that of the corresponding sphere circumscribing the tetrahedron formed 
by the four spheres, analogous to the formula (2p=P), connecting the radius of the 
nme-points circle of a triangle with its circumcircle. 
Spheres cutting Four Spheres at given Angles. —§§ 217-219. 
217. Let the given spheres be denoted by (1, 2, 3, 4), and let their orthogonal sphere 
be denoted by the symbol (5). Let x denote any sphere cutting (1, 2, 3, 4) at the 
angles 0, <£, if/, y; and let x cut (5) at the angle w. 
4 f 2 
