PROGRESS IN NINETEENTH CENTURY 39 



attention of Rayleigh, it is chiefly on the experimental side that the 

 subject has been enriched, as, for instance, by the labors of Langley 

 (1891) and Lilienthal. Langley (1897) has, indeed, constructed a 

 steam-propelled aeroplane which flew successfully; but man himself 

 has not yet flown. 



Moreover, the meteorological applications of aerodynamics con- 

 tained in the profound researches of Guldberg and Mohn (1877), 

 Ferrel (1877), Oberbeck (1882, 1886), Helmholtz (1888, 1889), and 

 others, as well as in such investigations as Sprung's (1880) on the in- 

 ertia.path, are as yet rather qualitative in their bearing on the actual 

 motions of the atmosphere. The marked progress of meteorology is 

 observational in character. 



Acoustics 



Early in the century the velocity of sound given in a famous equa- 

 tion of Newton was corrected to agree with observation by Laplace* 

 (1816). 



The great problems in acoustics are addressed in part to the elas- 

 tician, in part to the physiologist. In the former case the work of 

 Rayleigh (1877) has described the present stage of development, 

 interpreting and enriching almost every part discussed. In the latter 

 case Helmholtz (1863) has devoted his immense powers to a like 

 purpose and with like success. Konig has been prominently con- 

 cerned with the construction of accurate acoustic apparatus. 



It is interesting to note that the differential equation representing 

 the vibration of strings was the first to be integrated; that it passed 

 from D'Alembert (1747) successively to Euler (1779), Bernoulli 

 (1753) and Lagrange (1759). With the introduction of Fourier's series 

 (1807) and of spherical harmonics at the very beginning of the cen- 

 tury, D'Alembert's and the other corresponding equations in acous- 

 tics readily yielded to rigorous analysis. Rayleigh's first six chapters 

 summarize the results for one and for two degrees of freedom. 



Flexural vibration in rods, membranes, and plates become pro- 

 minent in the unique investigations of Chladni (1787, 1796, Akustik, 

 1802). The behavior of vibrating rods has been developed by Euler 

 (1779), Cauchy (1827), Poisson (1833), Strehlke (1833), Lissajous 

 (1833), Seebeck (1849), and is summarized in the seventh and eighth 

 chapters of Rayleigh's book. The transverse vibration of membranes 

 engaged the attention of Poisson (1829). Round membranes were 

 rigorously treated by Kirchhoff (1850) and by Clebsch (1862); ellip- 

 tic membranes by Mathieu (1868). The problem of vibrating plates 

 presents formidable difficulties resulting not only from the edge con- 

 ditions, but from the underlying differential equation of the fourth 

 degree due to Sophie Germain (1810) and to Lagrange (1811). The 



