705 
D, E 
j ustj 
the 
from 
■e ,S Y 
the 
(viz. 
:ircle 
•ibed 
like 
ouch 
lared 
distance of centres, or what is the same thing, squared difference of radii + 4 times 
the product of radii = squared distance of centres; that is, 
cC 2 e 2 (cos u — cos v) 2 + 4 (k — ae cos u) (k — ae cos v) = a 2 (cos u — cos v) 2 + b 2 (sin u — sin vf, 
or 
2 (k — ae cos u) (k — ae cos v) = b 2 {1 — cos (u — v)}. 
If for a moment we write tan \u = x, tan \v = y, and therefore 
1 — y 2 . 2x . 2y 
cos u = =- . cos v = -—^, sm u = , , sinii= ,—, 
1 + X 2 ' 1 +y 2 1 + X 2 1 +y 2 
. (1 — x 2 ) (1 — y 2 ) + 4xy . . 2 (x — y ) 2 
V (1 + X 2 ) (1 + y 2 ) V ' (1 + OC“) (1 + y 2 ) 
we have 
(, ae (1 — x?)) [, ae{\—y 2 )\ 
rVlrtt#-}“ 
(l+tf 2 )(l+y 2 y 
{k — ae + (k + ae) x 2 } [k — ae + (k + ae) y 2 } — b 2 (x — y) 2 , 
which is readily identified with the circular relation 
tan- „ - tan- * (*±“Y = tan- ." , ,, 
J \k - ae) \k - ae) \a 2 - k 2 
or, what is the same thing, in order that the circles U, V may touch, the relation 
between the parameters u, v must be 
tan -1 ■! ( 1 tan kvV — tan' 
Considering in like manner a circle, centre the point W, coordinates (acosw, 6sinw), 
and radius k — ae cos w, say the circle W; this will, as before, touch the circles R, S; 
and we may make W touch each of the circles U, V; viz. we must have 
'k + ae\* 
'k + ae\ì 
tan -1 ■{( " 1 ) tan— tan -1 \( 7 — ) tan -1 } = tan -1 
tan -1 ■{ I ^ + Cl ~\ tan 4til — tan -1 {(^-— a6 \ tan -1 kw\ = tan 
k 2 — a 2 e 2 \$ 
k 2 — a 2 e 2 \* 
7 / y. w i vvv “ ' « f 
where, in the last equation, tan -1 tan mus f be considered as denoting its 
value in the first equation increased by ir. Hence, adding the three equations, we have 
'k 2 — a 2 erV 
7T = 3 tan -1 
that is