158 
ON THE PROBLEM OF TACTIONS. 
[919 
91 
or putting m — n=2k, we have an 
f + it) - k 2 = k \/{(f - l) 2 +(v~ *) 2 }> 
and thence 
k 2 (f 2 + v 2 ) ~ (f + iv) 2 = k*, an 
for the equation of the conic. The last preceding equation gives 
V{((? “ l) 2 +(v~ ¿) 2 } =-k + . =-\{m-n) + ^pp, I 
or say 
£ + iv an ' 
V{(£- l) 2 + (v- 0 2 } - n = - i (m + w) + ; 
and we have similarly 
V{(£ +1) 2 + (v + if] -rn = -\(m + n)+ ^ + J V . viz 
wh 
This being so, it at once appears that, if (£, rj) are coordinates of a point on ^j s 
the conic, then the circle 
0 - f ) 2 + {y - v)° = a 2 , 
where ex] 
1 / \ £ + iv 
a = — £ (ra + ?i) + —, 
is a circle touching each of the given circles Z x , Z 2 . In fact, the distance of the 5 
centre from the point Z x is V{(£ +l) 2 + (v + t) 2 }, which is = a + ra, the sum of the two 
radii; and similarly the distance of the centre from the point Z 2 is \Z{(f — l) 2 + (77 — i) 2 }, 
which is = a + n, the sum of the two radii. 
Hence if (£', 77'), (£", 77") belong to any other two points on the conic, and we v [ z 
write 
a = — ^ (m + n)+ ^—p , 
the 
£ : 
b =-\{m + n)+ ^pp~ , 
anc 
c = — %(m + n) + , 
res 
we have 
(x - % ) 2 + (y - 77 ) 2 = a 2 , 
(X - f ) 2 + (y - 7)' ) 2 = b 2 , 
0 - D 2 + (g- v") 2 = c 2 , 
the 
for the equations of 
Writing as before /, g, 
three circles A, B, C each touching the two circles Z x , Z 2 . 
h for the mutual distances BC, CA, AB of the centres of these 
eqr 
circles, then 
P = (?-n 2 + W-v"y, 
or 
and similarly for g 2 and h 2 . But we have 
*-c = |l (r-n+ »•(>»'->!")). 
the 
*1,