I considered all the living which walk under the sun, with the second child that shall stand up in his stead.
I considered all the living - all the present generation of subjects. With the second child that shall stand up in his stead - in relation to ("with") the "second youth" (the legitimate successor of "the old king," as opposed to the "poor youth," the one first spoken of, about to be raised from poverty to a throne) - i:e., Rehoboam.
In his stead - the old king's.