MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
593 
lienee 
1 
Tpp.i 
1 p',i 
—+ J -+ — 
tt'q.s ’Tq-.s ^E.s 
1112 
—+—+ + 
R',3 
r S,4 TTS',4 
:0;. . . (248) 
or the sum of the reciprocals of the squares of the tangents from the points of 
intersection of four spheres to the spheres is equal to the reciprocal of the square of 
the radius of the sphere which cuts them orthogonally. 
Again, from equation (228), 
"J.l | 11 2 | 
1 - r .3 
'x,i 
1 P.l 
77, 
Q.x 
^8,4 7r *,4 
i; 
or if (x) cut (1, 2, 3, 4) at angles fa, fa, fa, fa, we have 
1 , 1 2 r, cos <A , 2 r„ cos <£, , 2 r, cos cb., . 2 r, cos <b, 
-+ - sec a; = - —+ y2 +—- a3 +— 4 -- 4 
P ^P.l *^"e, 3 ' 7r s,4 
(249) 
The Spheres which Touch Four given Spheres. —§§ 222, 223. 
222. Let the given spheres be denoted by (1, 2, 3, 4), their orthogonal sphere 
by (5). Then x denoting any sphere which touches them, we shall have by the 
equation 
sin 3 6i, 
^ 77 2,2) 
v Ar 3= 
V 7 77 4i4 , 
by taking the expression \ // 7r h l5 v 7 77 £i 2 , &c., with different signs, we obtain the 
eight values of sin 2 6i corresponding to the eight pairs of tangent spheres. 
The radii are given at once by the formula 
hr, 1, 2, 3, 4, 5 N 
U 1,2, 3, 4, 5, 
77 ”l, 15 
V 7 77o, o, 
^ 77 3,3’ 
v 774 , 4 , 
’L.n 
7r 1,2J 
7r l,33 
^1,4 
Aim 
71 % 23 
77o,, 3 ) 
^2,4 
As, u 
77 3,2’ 
7T 3 .37 
TLs.r 
77 -U 1) 
77 i,-2> 
3’ 
774,4 
= 0; 
(250) 
Thus, if p be the radius of the sphere touching (1, 2, 3, 4) externally, and 6i be 
the angle at which it cuts the orthogonal sphere, we have 
MDCCCLXXXVI. 4 G 
