Difference between research and résearch (with an accented e)

I suspect Wilson was using the accent mark as a stress indicator. The implied context, I believe, is that there had been a time when the noun "research" was normally stressed on the second syllable. Over some period of time before 1931 (the year in which Wilson wrote this), people began stressing the word on the first syllable. It's like referring, today, to someone "who lived in the days before dungarees had become jeans and pocketbooks had become purses".

'Résearch' is a nonce word created specifically for that one occasion to communicate a new idea. Sometimes nonce words become neologisms and become used by others. Sometimes not.

The accent on the 'e' is not a native English spelling. It is presumably intended by the author to change the more modest sounding 'research', to give the feel of a French word, which in English writing has a connotation of higher class or fanciness or highborn.

Searching google books for occurrences of 'résearch' finds nothing:

Likewise, searching French sources finds no evidence of a French word. The actual French translation of 'research' is 'recherches'.

Just because google ngrams doesn't find anything doesn't mean it doesn't exist (it's not searching web pages). But it's a good indication that it is either rare or not accepted by most people as a repeatable word.

To my understanding, the writer means that Gibbs, who died in 1903, did research when there was very little history of physics research in the United States, before the physics revolution (relativity and quantum mechanics) of the early twentieth century, before scientific research became a full-fledged salaried career, before there existed a competitive job market for academic research, when scientific research had more in common with the natural philosophy of the 1600s and 1700s than it did with the modern-day research-industrial complex funded by government and private industry, and when science was still mostly a leisure pursuit of individual scholars rather than a professional endeavor of large teams.

The accent mark and italicization stands for an affected pronunciation that carries connotations of pretension, as opposed to the humbler, less self-referential enterprise of earlier generations.

The writer, Edwin Bidwell Wilson, wrote this in the Bulletin of the American Mathematical Society, to an audience likely to understand how the discipline of scientific research was changing at the beginning of the twentieth century. I don't think it's fair to say it's bad writing.