The major breakthrough in this process was to send expanded cooled gas through a heat exchanger to cool incoming compressed gases, allowing for far lower temperatures.  
It avoids the central problem of the earlier Siemens process by using a throttling process rather than expansion through a turbine, thereby creating a system without moving parts exposed to ultralow temperatures.  If the system is modified to send a portion of the vapor in a superheated state through an expander instead of the Joule-Thomson valve, it becomes known as the Claude process.  The major breakthrough in this process was to send expanded cooled gas through a heat exchanger to cool incoming compressed gases, allowing for far lower temperatures.  FTP, what is this refrigeration process patented in 1895 that is the most common method of liquefying gases?
It avoids the central problem of the earlier Siemens process by using a throttling process rather than expansion through a turbine, thereby creating a system without moving parts exposed to ultralow temperatures.  
Hampson-Linde cycle