Orthogonal Cauchy-like matrix

Please give a look at this short note.

Yes, here is an example for n = 3.

r' =  [-1.085216443606418   4.526028779191116  -0.111128247133696]
s' =  [0.552760089055250    0.079464242975571  -0.006871962674798]
x' =  [-2.748286685551109   1.237373231951619  -1.153274317177488]
y' =  [ 3.750107821687254  -2.661680052752817  -1.152510574989135]
C =
   [0.092309621606373   0.995719385103790  -0.004673316541826
   -0.995651542853765   0.092242250711209  -0.013014315066420
    0.012527528559379  -0.005854341324361  -0.999904389287486]

C*C' =  
   [1.000000000000081   0.000000000000002   0.000000000000001
    0.000000000000002   1.000000000000000   0.000000000000003
    0.000000000000001   0.000000000000003   1.000000000000528]