PROGRESS IN NINETEENTH CENTURY 37 



the century in the treatment of waves, of discontinuous fluid motion, 

 and in the dynamics of solids suspended in frictionless liquids. 

 Kelland (1844), Scott Russel (1844), and Green (1837) dealt with 

 the motion of progressive waves in relatively shallow vessels, Ger- 

 ster (1804) and Rankine (1863) with progressive waves in deep water, 

 while Stokes (1846, 1847, 1880), after digesting the contemporaneous 

 advances in hydrodynamics, brought his powerful mind to bear 

 on most of the outstanding difficulties. Kelvin introduced the case 

 of ripples (1871), afterwards treated by Rayleigh (1883). The soli- 

 tary wave of Russel occupied Boussinesq (1872, 1882), Rayleigh 

 (1876), and others; group-waves were treated by Reynolds (1877) 

 and Rayleigh (1879). Finally the theory of stationary waves re- 

 ceived extended attention in the writings of de St. Venant (1871), 

 Kirchhoff (1879), and Greenhill (1887). Early experimental guid- 

 ance was given by the classic researches of C. H. and W. Weber 

 (1825). 



The occurrence of discontinuous variation of velocity within the 

 liquid was first fully appreciated by Helmholtz (1868), later by 

 Kirchhoff (1869), Rayleigh (1876), Voigt (1885), and others. It lends 

 itself well to conformal representations. 



The motions of solids within a liquid have fascinated many inves- 

 tigators, and it is chiefly in connection with this subject that the 

 method of sources and sinks was developed by English mathema- 

 ticians, following Kelvin's method (1856) for the flow of heat. The 

 problem of the sphere was solved more or less completely by Poisson 

 (1832), Stokes (1843), Dirichlet (1852); the problem of the ellip- 

 soid by Green (1833),Clebsch (1858), generalized by Kirchhoff (1869). 

 Rankine treated the translatory motion of cylinders and ellipsoids 

 in a way bearing on the resistance of ships. Stokes (1843) and Kirch- 

 hoff entertain the question of more than one body. The motion 

 of rings has occupied Kirchhoff (1869), Boltzmann (1871), Kelvin 

 (1871), Bjerknes (1879), and others. The results of C. A. Bjerknes 

 (1868) on the fields of hydrodynamic force surrounding spheres, 

 pulsating or oscillating, in translatory or rotational motion, accent- 

 uate the remarkable similarity of these fields with the corresponding 

 cases in electricity and magnetism, and have been edited in a unique 

 monograph (1900) by his son. In a special category belong certain 

 powerful researches with a practical bearing, such as the modern 

 treatment of ballistics by Greenhill and of the ship propeller of 

 Ressel (1826), summarized by Gerlach (1885, 1886). 



The numerous contributions of Kelvin (1888, 1889) in particular 

 have thrown new light on the difficult but exceedingly important 

 question of the stability of fluid motion. 



The century, moreover, has extended the working theory of the 



