September 22, 1905.] 



SCIENCE. 



361 



and enriching almost every part discussed. 

 In the latter case Helmholtz (1863) has de- 

 voted his immense powers to a like purpose 

 and with like success. Konig has been 

 prominently concerned with the construc- 

 tion of accurate acoustic apparatus. 



It is interesting to note that the differ- 

 ential 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 begin- 

 ning of the century, D'Alembert 's and the 

 other corresponding equations in acoustics 

 readily yielded to rigorous analysis. Ray- 

 leigh's first six chapters summarize the re- 

 sults for one and for two degrees of free- 

 dom. 



FJexural vibration in rods, membranes 

 and plates become prominent in the unique 

 investigations of Chladni (1787, 1796, 

 'Akustik,' 1802). The behavior of vibrat- 

 ing 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 en- 

 gaged the attention of Poisson (1829). 

 Round membranes were rigorously treated 

 by Kirchhoff (1850) and by Clebsch 

 (1862) ; elliptic membranes by Mathieu 

 (1868). The problem of vibrating plates 

 presents formidable difficulties resulting 

 not only from the edge conditions, but from 

 the underlying differential equation of the 

 fourth degree due to Sophie Germain 

 (1810) and to Lagrange (1811). The so- 

 lutions have taxed the powers of Poisson 

 (1812, 1829), Cauchy (1829), Kirchhoff 

 (1850), Boussinesq (1871-79) and others. 

 For the circular plate Kirchhoff gave the 

 complete theory. Rayleigh systematized 

 the results for the quadratic plate and the 



general account makes up his ninth and 

 tenth chapters. 



Longitudinal vibrations which are of 

 particular importance in case of the organ 

 pipe, were considered in succession by 

 Poisson (1817), Hopkins (1838), Quet 

 (1855) ; but Helmholtz in his famous paper 

 of 1860 gave the first adequate theory of 

 the open organ pipe, involving viscosity. 

 Further extension was then added by 

 Kirchhoff (1868), and by Rayleigh (1870, 

 et seq.), including particularly powerful 

 analysis of resonance. The subject in its 

 entirety, including the allied treatment of 

 the resonator, completes the second volume 

 of Rayleigh's 'Sound.' 



On the other hand, the whole subject of 

 tone quality, of combination and difference 

 tones, of speech, of harmony, in its physical, 

 physiological and esthetic relations, has 

 been reconstructed, using all the work of 

 earlier investigators by Helmholtz (1862), 

 in his masterly ' Tonempfindungen. ' With 

 rare skill and devotion Konig contributed 

 a wealth of siren-like experimental appur- 

 tenances. 



Acousticians have been fertile in de- 

 vising ingenious methods and apparatus, 

 among which the tuning fork with reso- 

 nator of Marloye, the siren of Cagniard 

 de le Tour (1819), the Lissajous curves 

 (1857), the stroboscope of Plateau (1832), 

 the manometric flames , of Konig ( 1862, 

 1872), the dust methods of Chladni (1787) 

 and of Kundt (1865-68), Melde's vibrating 

 strings (I860, 1864), the phonograph of 

 Edison and of Bell (1877), are among the 

 more famous. 



HEAT : THERMOMETRY. 



The invention of the air thermometer 

 dates back at least to Amontons (1699), but 

 it was not until Rudberg (1837), and more 

 thoroughly Regnault (1841, et seq.) and 

 Magnus (1842) had completed their work 

 on the thermal expansion and compressi- 



