Does anyone know the threading size for the temperature switch?

A temperature switch turns things off and on such as cooling fans.
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A Transmission Temperature gauge sending unit sends the temp information to a gauge. They are so many different aftermarket brand and types of oil temp gauge kits that no one can tell you what thread size one will have.

Your next problem is finding a location the the Transmission to drill and tap a hole for the sending unit. The sending unit must be submerged in oil at all times to work correctly and give an accurate temperature reading. If the mower has a hydro filter that is attached to oil lines the temp gauge can be installed on one of the lines using a T.

Normally hydraulic oil is ok up to around 350*F . After it passes 350* for a period of time it begins to break down and and when it get to the point of being "burnt" you will know it by the smell.