Septembeb 22, 1905.] 



SCIENCE. 



359 



(1755, 1759) had left the hands of La- 

 grange (1788) in their present form. In 

 relatively recent time, H. Weber (1868) 

 transformed them in a way combining cer- 

 tain advantages of both, and another trans- 

 formation was undertaken by Clebsch 

 (1859). Hankel (1861) modified the 

 equation of continuity, and Svanberg and 

 Edlund (1847) the surface conditions. 



Helmholtz in his epoch-making paper of 

 1858 divided the subject into those classes 

 of motion (flow in tubes, streams, jets, 

 waves) for which a velocity potential ex- 

 ists and the vortex motions for which it 

 does not exist. This classification was car- 

 ried even into higher orders of motion by 

 Craig and by Rowland (1881). For cases 

 with a velocity potential, much progress 

 has been made during the century in the 

 treatment of waves, of discontinuous fluid 

 motion, and in the dynamics of solids sus- 

 pended in frictionless liquids. Kelland 

 (1844), Scott Russel (1844) and Green 

 (1837) dealt with the motion of progressive 

 waves in relatively shallow vessels, Gerster 

 (1804) and Rankine (1863) with progress- 

 ive weaves in deep water, while Stokes 

 (1846, 1847, 1880) after digesting the co- 

 temporaneous advances in hydrodynamics, 

 brought his powerful mind to bear on most 

 of the outstanding difficulties. Kelvin in- 

 troduced the case of ripples (1871), after- 

 wards treated by Rayleigh (1883). The 

 solitary 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 received ex- 

 tended attention in the writings of de St. 

 Venant (1871), Kirchhoff (1879) and 

 Greenhill (1887). Early experimental 

 guidance was given by the classic re- 

 searches of C. H. and W. Weber (1825). 



The occurrence of discontinuous varia- 

 tion 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 representa- 

 tions. 



The motions of solids within a liquid 

 have fascinated many investigators and it 

 is chiefly in connection with this subject 

 that the method of sources and sinks was 

 developed by English mathematicians, fol- 

 lowing 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 ellipsoid by 

 Green (1833), Clebsch (1858), generalized 

 by Kirchhoff (1869). Rankine treated' 

 the translatory motion of cylinders and 

 ellipsoids in a way bearing on the re- 

 sistance of ships. Stokes (1843) and 

 Kirchhoff 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 hydrodynamie force 

 surrounding spheres, pulsating or oscillat- 

 ing, in translatory or rotational motion ac- 

 centuate the remarkable similarity of these 

 fields with the corresponding cases in elec- 

 tricity and magnetism and have been edited 

 in a unique monograph (1900) by his son. 

 In a special category belong certain power- 

 ful 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 tides due to Newton 

 (1687) and Laplace (1774), through the 

 labors of Airy, Kelvin and Darwin. 



