for any segment of the rope, at an angle 'a' to the starting radius perpendicular to T1,
we have dT=dF
where dT is the slight increment in the tension, for an increment of a by da,
we have dF=kdN=dT
we also have dN=Tda
so kTda=dT
or Kda=dT/T
\int_{0}^{A}{da}= 1/K \int_{T1}^{T2}{dT/T}
so AK=log(T2/T1)
or T2=T1e^kA
cheers!!