Review paper: What Kelvin might have written about Elasticity

One hundred and fifty five years ago, Kelvin published the first part of a fundamental analysis of the elastic tensor, in which he proposed a coordinate-free representation through its eigensystem. His thoughts were apparently far ahead of his time, since it took 125 years before the paper elicited a positive reaction (it is now accessible through several modern reviews). Science not only lost track for 125 years of the original paper but also lost the ideas Kelvin might have proposed in the second part, a publication that was never put to paper, presumably in view of the lack of appreciation of the first part. In an attempt to establish what might have been on Kelvin's mind for a second part, one has to forget the progress of mathematical physics in the intervening time and base all arguments strictly on the content of the first part and on the state of science in the second half of the 19thcentury. The theory of elasticity would certainly have developed faster, had Kelvin's paper peen appreciated by his peers. But a theory based on Kelvin's ideas would be fruitful even today.