You are using a combo wireless router/modem which the desktop with wireless adapter is connecting. And so the other desktop is wired to the combo router. Since there is connection established on the wireless desktop and suddenly drops, the first thing to look here is distance and inteference. Distance - Consider moving first atleast in one room the wireless Desktop to confirm the problem.
Interference - Isolate the Desktop from any type of electronic appliances. Radio signal, Cordless phone, Microwave oven.
Consider using/reading the manual that came with your router in any case that your connection is not secured. Also, please provide all protection softwares installed such as firewall, antivirus, spyware,scanners.