566 
PROBLEMS AND SOLUTIONS. 
[485 
exterior to each other. The foregoing equations signify that 90° — a, 90° — /3 are the 
inclinations to the line of centres of the inverse and the direct common tangents 
respectively, and that m is the length of the inverse common tangent. And the 
theorem is, that considering two circles as above, and taking M a variable point in 
X' 
G 
the line of centres, if r, s denote the tangential distances of M from the two circles 
respectively, and if m be the length of the inverse common tangent of the two 
circles, then the angles 9, <j) determined by the equations 
m sin 0 = r + s, m sin (¡3 = r — s, 
are connected by the relation 
cos ¡3 = cos 6 cos </> + sin 6 sin </> cos a, 
(a, /3) being constant angles, determined as above. 
7. 
It is to be remarked that, assuming 
sin a _ V{4c 2 — (a + 5) 2 } 
sin /3 V {4c 2 - (a — 6j 2 } ’ 
that is, k = inverse common tangent 4- direct common tangent, then we have 
cos a = V(1 — k 2 sin 2 /3) = A/3, 
or the equation in 9, cf) becomes 
cos /3 = cos 0 cos <f> + sin 0 sin (f) A/3, 
which is the algebraical equation connecting the amplitudes of the elliptic functions 
in the relation F (0) + F (d>) = F (¡3). 
8. It is very noticeable that the above figure leads to another relation in elliptic 
functions, viz. it is the very figure employed for that purpose by Jacobi; in fact, 
considering therein YM as a variable tangent meeting the circle A in the two points 
X and X', then if 2\{r, 2-»// denote the angles GAX, GAX' respectively, it is easy to 
see geometrically that we have dyjr : dyjr' = YX : YX'; where 
(YX) 2 = (BX f — b 2 , — 4c 2 + a 2 + 4ac cos 2yjr — b 2 , = (2c + a) 2 — b 2 — 8ac sin 2 yjr, 
and similarly (YX') 2 = (2c + a) 2 — b 2 — 8ac sin' 2 \Jr', that is, writing 
differential equation is 
dijr d-yjr' _ 
V(1 — l 2 sin 2 l/r) \/(l — l 2 sin 2 -yfr') 
8 ac 
(2c + a) 2 — b 2 ’ 
the