We manually solve the equations when the spheres have centers a e_2, -e_2, and pm e_j for j=3,...,n Here are the equations V^2*(P^2 \pm 2pj+1-r^2)-vj^2 for j=3,...,n V^2*(P^2 - 2ap2+a^2-r^2)-v2^2 V^2*(P^2 + 2p2+1-r^2)-v2^2 P cdot V ################################################################### We have the 2n-4 equations for the spheres with centers at \pm e_j 2 2 2 2 V (P - 2 pj + 1 - r ) - vj 2 2 2 2 V (P + 2 pj + 1 - r ) - vj Combining these, we obtain p3 = p4 = ... = pn = 0, v3^2 = v4^2 = ... = vn^2 = 0, and the equation from the spheres with center pm e_3: 2 2 2 2 v3 - V (P + 1 - r ) ################################################################ The equations for the spheres with centers at ae_2 and -e_2 are: 2 2 2 2 2 2 f := V (P - 2 a p2 + a - r ) - a v2 2 2 2 2 g := V (P + 2 p2 + 1 - r ) - v2 Consider f+a*g: 2 2 2 2 2 2 -(1 + a) (-a V + a v2 - V P + V r ) and also f-a^2*g: 2 2 2 2 2 V (1 + a) (-a P - 2 a p2 + a r + P - r ) Using V^2\neq 0 and a\neq -1, (which allow us to cancel factors) and previous equations, (so that P^2=p1^2+p2^2) we obtain the following system of equations: p3 = p4 = ... = pn = 0 v3^2 = v4^2 = ... = vn^2 = 0 w := p1 v1 + p2 v2 2 2 2 x := (1 - a) (p1 + p2 - r ) - 2 a p2 2 2 2 2 2 y := V (p1 + p2 - r + a) - a v2 2 2 2 2 2 z := v3 - V (p1 + p2 - r + 1) We combine the second and third equations to solve for p2: (V - v2) (V + v2) (a - 1) P2 := 1/2 ------------------------- 2 V Plugging into the first equation, we solve for p1: (V - v2) (V + v2) (a - 1) v2 P1 := - 1/2 ---------------------------- 2 V v1 We substitute these into the second equation and clear the denominator to obtain a sextic. We use V^2=v1^2+v2^2+(n-2)v3^2 2 2 6 2 2 4 2 W := (a - 4 r + 2 a + 1) v1 + (-8 r + a + 1 + 2 a) v1 v2 2 2 2 2 4 2 + (2 a n + 16 r - 4 + 4 a n - 4 a - 8 a - 8 r n + 2 n) v1 v3 2 2 4 2 2 2 2 2 2 2 - 4 r v1 v2 + (-4 + 16 r - 4 a + 2 a n + 2 n - 8 r n) v1 v2 v3 2 2 2 2 2 2 2 2 + (-4 r n + 8 a + a n - 4 a n + 4 - 8 a n + 4 a - 4 n + 2 a n + n 2 2 2 4 - 16 r + 16 r n) v1 v3 2 2 2 2 2 2 2 4 + (-8 a - 4 a n + 4 a + 8 a n + 4 + n - 2 a n - 4 n + a n ) v2 v3 Here is the formula in the paper: 2 2 2 2 2 2 2 4 2 2 2 2 2 X := (1 - a) (v1 + v2 ) (V - v2 ) - 4 r V v1 + 4 a v1 V (V - v2 ) We check this: simplify(subs(V=sqrt(v1^2+v2^2+(n-2)*v3^2),W-X)) 0 Adding the third equation and the fourth equation, z+y, we obtain 2 2 2 Y := (a - 1) v1 - v2 + (3 + a n - n - 2 a) v3 We substitute for the squared variables in this equation to obtain (a - 1) alpha - beta + (3 + a n - n - 2 a) gamma Solving for beta and plugging into the sextic we obtain the cubic 2 2 3 3 2 2 2 (n - 2) (a - 1) (3 + a n - n - 2 a) gamma - (12 a n - 16 r a + 16 r a 2 2 2 2 2 2 2 2 2 + 28 a n + 20 a + 8 r a n + 4 r a n - 16 r a n + 4 a n + 12 2 3 2 2 2 2 3 - 18 a n - 3 a n - 10 n - 7 a n + 4 r + 2 n - 12 a - 28 a) alpha 2 2 2 3 2 2 2 3 gamma - (8 a + 8 r a n + 6 a + n - 3 - 16 r a + 8 r a - 3 a n 2 2 2 2 2 3 - 5 a n + a - a n) alpha gamma - a (-a + 4 r a - 2 a - 1) alpha We compute the discriminant of this cubic, discrim(subs(gamma=1,Xx),alpha) This discriminant is quite large, and it does not factor > nops(Disc); 116 > degree(Disc); 16 > nops(factor(Disc)); 116 > interface(quiet=true): We consider the leading and trailing coefficients of the cubic Coefficent of gamma^3: 2 2 (n - 2) (a - 1) (3 + a n - n - 2 a) Coefficent of alpha^3: 2 2 -a (-a + 4 r a - 2 a - 1) This shows that for each n, neither alpha nor gamma vanishes for general a, r. To check for beta, we solve the linear equation for alpha and plug that into the cubic 2 2 3 2 3 2 2 2 -a (-a + 4 r a - 2 a - 1) beta + (4 r a n - 3 a n - 4 r a n + 7 a n 2 2 2 3 3 2 3 2 2 + 12 r a - 19 a - 5 a n + 8 a + n - 8 r a - 3 + 2 a + 8 r a) beta 3 2 2 2 2 2 2 2 gamma + (-11 a - 8 r a n + 16 a - 4 r + 2 a n + 8 r a n + 16 r a 2 3 2 2 + a - 2 n - 6 a n + 6 a n - 24 r a + 6) beta gamma 3 + (2 r + 1 + a) (2 r - 1 - a) (3 + a n - n - 2 a) gamma We consider the leading and trailing coefficients of this cubic Coefficent of gamma^3: (2 r + 1 + a) (2 r - 1 - a) (3 + a n - n - 2 a) Coefficent of beta^3: 2 2 -a (-a + 4 r a - 2 a - 1) This shows that for each n, neither beta nor gamma vanishes for general a, r.