566 
[705 
705. 
PROBLEMS AND SOLUTIONS. 
[From the Mathematical Questions ivith their Solutions from the Educational Times, 
vols. xiy. to lxi. (1871—1894).] 
[Vol. xiv., July to December, 1870, pp. 17—19.] 
3002. (Proposed by Matthew Collins, B.A.)—If every two of five circles A, B,C, D, E 
touch each other, except D and E, prove that the common tangent of D and E is just 
twice as long as it would be if D and E touched each other. 
Solution by Professor Cayley. 
Consider the ellipse ^-+^r = l, foci R, S: the coordinates of a point U on the 
a 2 0- 
ellipse may be taken to be (a cos u, b sin u), and then the distances of this point from 
the foci will be 
r = a (1 — e cos u), s = a (1 + e cos u). 
Taking k arbitrarily, with centre R describe a circle radius a —k, with centre S 
a circle radius a + k, and with centre U a circle radius k — ae cos u: say these are the 
circles R, S, TJ respectively; the circle U will touch each of the circles R, S (viz. 
assuming ae<k<a, so that the foregoing radii are all positive, it will touch the circle 
R externally and the circle S internally). 
Considering next a point V, coordinates (a cos v, b sin v), and the circle described 
about this point with the radius k — ae cos v, say the circle V; this will touch in like 
manner the circles R, S respectively. And the circles TJ, V may be made to touch 
each other externally; viz. this will be the case if squared sum of radii = squared