I know some of this has appeared in math-fun before, but don't recall the final generalisation making a bow at the time --- apologies if it's old stuff! There is a very neat proof in terms of Cayley-Menger type determinants ("distance geometry"). Quoted (more or less) from Pat Ballew at URL www.pballew.net/soddy.html << The Kiss Precise by Frederick Soddy For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. Four circles to the kissing come. The smaller are the benter. The bend is just the inverse of The distance from the centre. Though their intrigue left Euclid dumb There's now no need for rule of thumb. Since zero bend's a dead straight line And concave bends have minus sign, The sum of the squares of all four bends Is half the square of their sum. To spy out spherical affairs An oscular surveyor Might find the task laborious, The sphere is much the gayer, And now besides the pair of pairs A fifth sphere in the kissing shares. Yet, signs and zero as before, For each to kiss the other four The square of the sum of all five bends Is thrice the sum of their squares. In _Nature_, June 20, 1936 Later another verse was written by Thorold Gosset to describe the even more general case in n dimensions for n+2 hyperspheres of the n-th dimension. The Kiss Precise (Generalised) by Thorold Gosset And let us not confine our cares To simple circles, planes and spheres, But rise to hyper flats and bends Where kissing multiple appears, In n-ic space the kissing pairs Are hyperspheres, and Truth declares - As n + 2 such osculate Each with an n + 1 fold mate The square of the sum of all the bends Is n times the sum of their squares. In _Nature_ January 9, 1937.
While I can't claim to have published the next result, I did in fact discover it independently. So I feel justified in attempting an ode in the traditional vein. The Kiss Precise (Further Generalised) by Fred Lunnon How frightfully pedestrian My predecessors were To pose in space Euclidean Each fraternising sphere! Let Gauss' k squared be positive When space becomes elliptic, And conversely turn negative For spaces hyperbolic: Squared sum of bends is sum times n Of twice k squared plus squares of bends. In more conventional notation, let n+2 spheres be mutually tangent at distinct points in uniform space of n dimensions with (Gaussian, second) curvature k^2. The Descartes / Lachlan / Soddy/ Gosset relation between their curvatures x_i generalises to (\sum_i x_i)^2 = n (\sum_i x_i^2 + 2 k^2) . Examples: 4 circles touching at 6 points in plane n = 2: Euclidean k^2 = 0: x = [-1,2,2,3], 36 = 2(18 + 0); spherical k^2 = 1: x = [0,1,1,2], 16 = 2(6 + 2); hyperbolic k^2 = -1: x = [7,1,1,1], 100 = 2(52 - 2). See Ivars Petersen "Circle Game" in Science News (2001) \bf 159 (16) p.254 http://ww.sciencenews.org/articles/20010421/bob18.asp Result (rather obscurely) implicit in Lagarias, Mallows, Wilks \sl Beyond the Descartes Circle Theorem \rm, Amer. Math. Monthly also at http://xxx.lanl.gov/abs/math.MG/0101066. [apollo/0101066.pdf] Graham et al \sl Apollonian Circle Packings: Number Theory \rm http://xxx.lanl.gov/abs/math.NT/0009113. Websites: Graham at http://math.ucsd.edu/~fan/ron Lagarias, Wilks at http://www.research.att.com/~jcl, ~alan Fred Lunnon