If we assume the lowering of the temperature of maximum density to be proportional to the pressure, which is the simplest and most natural hypothesis, we may write

where p is in tons-weight per square inch.

Now Thomson's thermodynamic result is of the form

This becomes, with our assumption,

As the left-hand member is always very small, no sensible error will result from integrating on the assumption that t is constant on the right (except when the quantity in brackets is very small, and then the error is of no consequence).