A signed copy of one of Stokes's most important mathematical contributions (1848)

STOKES, George Gabriel

On the critical values of the sums of periodic series. From the Transactions of the Cambridge Philosophical Society. Vol. VIII. Part V.

Published: Cambridge, John W. Parker

Date: 1848

4to (275 × 215 mm), pp. 53, [1, blank]; a largely, clean, crisp text, one or two light smudges; inscribed by the author to 'Rev. (?) W. B. Hopkins'; very good in contemporary brown paper wrappers, a paper label with title in the author's hand, to upper cover, a little frayed at edges with a couple of small holes to the upper cover.

Provenance: William Hopkins (1793-1866) was Stokes’s mathematics tutor at Cambridge – he was indeed so famous as a successful tutor that he earned the sobriquet ‘senior wrangler maker’. As well as Stokes he tutored Lord Kelvin, Arthur Cayley, James Clerk Maxwell, and Isaac Todhunter.

COPAC records three UK copies at Cambridge, the National Maritime Museum and Senate House.

Original offprint from the Transactions of the Cambridge Philosophical Society, VIII, 5 (1848): 533-583. Senior Wrangler and Smith’s Prizeman in 1841, Stokes was encouraged by the example of George Green to venture into original research in applied mathematics. The first field he worked in was hydrodynamics, and he published On the steady motion of incompressible fluids in 1842, followed by On the theories of the internal friction of fluids in motion, 1845. At about the same time he also started to work in optics, gravitation and a number of other fields in the physical sciences. Nevertheless his interest in pure mathematics continued throughout the same period. In the present work Stokes contributed to the mathematics of infinite series. ‘The notion of uniform convergence of a series ∑1∞ un (x) requires that given any ε, there exists an N such that for all n > N, |S (x) – ∑1n un (x)| < ε for all x in some interval. S (x) is of course the sum of the series. This notion was recognized in and for itself by Stokes, a leading mathematical physicist [in the present work], and independently by Philipp L. Seidel (1821-96). Neither man gave the precise formulation. Rather both showed that if a sum of a series of continuous functions is discontinuous at x0 then there are values of x near x0 for which the series converges arbitrarily slowly. Also neither related the need for uniform convergence to the justification of integrating a series term by term. In fact, Stokes accepted Cauchy’s use of term-by-term integration. Cauchy ultimately recognized the need for uniform convergence in order to assert the continuity of the sum of a series of continuous functions but even he at that time did not see the error in his use of term-by-term integration of series.’ (Kline, p. 965)

Very good