A charge $Q$ is located at a distance $r>R$ from the center of a non-grounded conducting spherical shell with radius $R$ and total charge $q_s$. The field external to the shell can be mimicked by the combination of the image charge plus a second image charge. What is this second charge, and where is it located?
It's problem $3.16$ in Electricity and Magnetism by Purcell.
Purcell argued Since the image charges produce the same external field as the shell, Gauss's law implies that the charge on the actual shell is $-QR/r$, whereas we are told that the charge is $q_s$. We can remedy this by placing another image charge of $q_s+QR/r$ at the center. The total charge on the actual shell is now $q_s$, as desired. Furthermore, the boundary condition of constant potential on the shell is still satisfied, by symmetry, because the second image charge is located at the center. So by the uniqueness theorem, the field from our two image charges mimic the (external) field from the shell.
The external field is the same as produced by the charge $Q$ and $q=-QR/r$ for the grounded shell. I don't understand How gauss law implies the charge on the shell to be $-QR/r$? There wasn't any net charge on the shell initially. I also don't get how the boundary conditions are satisfied in the two problem?