 This is precisely what the first London equation states.To obtain the second equation, take the curl of the first London equation and apply Faraday's law,
to obtain

As it currently stands, this equation permits both constant and exponentially decaying solutions.The Londons recognized from the Meissner effect that constant nonzero solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression in the parentheses must be identically zero.