586 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
becomes 
0 , 
1 
^*2 
l 
To’ 
1 _ 
rJ 
which may be written 
or 
- 1 , 
1 , 
1 , 
1 , 
1 , 
1 
r 2 ’ 
1 , 
- 1 , 
1 , 
1 , 
1 , 
r, r. 
1 , 1 , 1 
1 , 
■ 1 , 
1 , 
1 , 
1 , 
1 , 
■ 1 , 
3, 
A ^ 1 
Y- -- Y— 
r,i 
1 ' 2 
: 0 ; 
1 , 11.1 
o— j- 2 + 2 -f— 2 )■—■ ( b ~ H , “by 
1 1 2 r S r i r 5 / Vl r l r S T i 1 b 
1 . 1 . 1 . 1 . 1 \ 3 
(235) 
216. If the spheres (1, 2, 3, 4, 5) have a common tangent sphere, which we will 
denote by ( x ), then the equation 
n h J’ 
V, 1 , 
2, 3, 4, 5\ 
2, 3, 4, 5/ 
= 0 , 
gives us 
D 
^ 77 1,1’ 
V 77 -’,Z> 
v 7 77 - 3,35 
n/ 77 - 5,5 
^ 77 i,\i 
7 r l, 1 » 
7T l,2’ 
7r i, 35 
7r l,45 
77 - 1,5 
\t TTo, 2j 
77 2, l; 
7T 2^ 0 , 
7r o ] 3 , 
77 - 3,45 
^ 2,5 
TT^g, 
^.n 
77 3, 2 s 
7 r 3,35 
77-3,45 
TT'S, 5 
V 7 
Tyi 
77 *4-, 25 
7 r l, 35 
TT-.i, 4 , 
77-4,5 
V 7 ^5,0> 
77 5,1’ 
77 5, 25 
77 o,3’ 
^5,5 
where the positive or negative sign is to be taken with any expression, such as \/tt„i,„, 
according as the spheres (m, n) have contact with (x) of the same or opposite kind. 
Now if we write 777 )i ;tv 7 7JV,r>77' M =C,/, in which t,- tS will denote the length of the 
common tangent of the two spheres (r, s ), “ direct ” if the positive sign is to be taken, 
“ indirect” if the negative sign is to be taken ; then we can deduce at once from this 
equation the equation of condition, due to Dr. Casey, 
