The Thermodynamic Properties and Molecular Structure of Cyclohexane, Methylcyclohexane, Ethycyclohexane and the Seven Dimethylcyclohexanes
[The question is actually what is the experimental energy difference between the boat and chair conformations of 1,3-dimethylcyclohexane - if this paper does not have the answer you need to find another source (textbooks cannot be used - must be a paper from a scientific journal with the measuremnts JCB]

While the above source is quite exhaustive in its research into cyclohexane, it might be a little long winded.

The following discussion should be easier to understand, and has more relevant information:
Cyclohexane Conformational Analysis

The most important information, relative to this subject is found near the end, where the specific energy confomations of trans-1,3-dimethylcyclohexane are found.

Important points to take away from both of these papers:

  • In the realm of 1,3-dimethylcyclohexane, the cis conformer is the most stable conformation
  • The trans-1,3-dimethylcyclohexane isomer has one methyl axial in both ring-flip conformers, so that it is less stable than the cis isomer.
  • There is another isomer of 1,4-dimethylcyclohexane, the cis isomer. In the cis isomer, here, one methyl group is equatorial and one axial, and ring flipping simply gives another equivalent conformation. Since one methyl is axial, this costs 1.8 kcal of steric strain.
  • In the cis isomer, one substituent is vertically up (or down) and the other slant up (or down). So they both point, in a general sense, in the same direction, i.e., both up or both down.