The rate of change of Tmax was approximated as the first difference of the time series. This produces data point offset by 0.5 months when correctly centred.
1928.083333 0.000000 1928.041667 42.800000
1928.166667 0.000000 1928.125000 59.200000
1928.250000 0.000000 1928.208333 96.400000
1928.333333 0.000000 1928.291667 183.600000
1928.416667 0.000000 1928.375000 249.600000
1928.500000 0.000000 1928.458333 258.300000
1928.583333 0.000000 1928.541667 105.700000
1928.666667 0.000000 1928.625000 173.000000
1928.750000 0.000000 1928.708333 145.200000
Since the diff amplifies the noise, sun-hours was taken for the abscissa. The plot and NNLS regression were done with a data lag of one interval on the temperature data, leaving a net lag of 0.5 months. ie temperature diff leads sun-hours.
This brings into question the idea of sun-hours having a causal link on Tmax. It rather suggests common causation. This can readily be suggested to be the annual increase in insolation both warming ground and air temps as well as affecting the weather systems encroaching from the North Atlantic.
There is some broadening in the cooler to mid parts of the cycle indicating some phase lag still present. The warmer part of the cycle appears to have negligible lag between the variables. Thus there is a small difference in the magnitude of the phase response across the annual cycle. Otherwise the relationship seems close to being linear.
[The title of the graph shows 1 mo lag. This is the lag of one line in the file. The correct analytical lag is 0.5mo as stated above.]
Similar processing with 1.5 mo lag is shown here: