We define the density `rho` in the frame K in such a way that `rhodxdydz` is the rest mass `dm_0` of the element. That is `rhodxdydz=rho_0dx_0dy_0dz_0`, where `rho_0` is the proper density `dx_0`, `dy_0`, `dz_0` are the dimensions of the element in the rest frame `K_0`. Now
`dy=dy_0`, `dz=dz_0`, `dx=dx_0sqrt(1-v^2/c^2)`
if the frame K is moving with velocity, v relative to the frame `K_0`. Thus
`rho=(rho_0)/(sqrt(1-v^2/c^2))`
Defining `eta` by `rho=rho_0(1+eta)`
We get `1+eta=(1)/(sqrt(1-v^2/c^2))` or , `v^2/c^2=1-(1)/((1+eta)^2)=(eta(2+eta))/((1+eta)^2)`
or `v=csqrt((eta(2+eta))/((1+eta)^2))=(csqrt(eta(2+eta)))/(1+eta)`