I have a TIFF file in the TM65 (Transverse Mercator) projection, and want to convert it to the OSGB (British National Grid) projection. I'm trying to do this in ArcGIS using the Project Raster tool.

I can select the input and output co-ordinate system fine, but it then asks me to choose a Geographic Transformation. It seems that I have to choose two - one to go from TM65 to WGS84, and then another to go from WGS84 to OSGB. My question is - which one should I choose? What are the differences between them? I have tried choosing some almost at random (and also tried matching them up - so I do TM65->WGS84_2 and then WGS84_2 to OSGB) and each time once the points are reprojected they are displaced from their true locations - the amount varies between fairly small (but large enough to be an issue) and quite large.

Is there any way I can get an accurate conversion between TM65 and OSGB? Am I using the wrong tool?