IIECENT PROGRESS IH HYDRODYNAMICS. 63 



than that of the fluid it is repelled or attracted according as the ratio of 

 its radius to the distance between the centres is less or greater than 

 1 _4/||(l + 2p)}. In the same year also, Bjerknes^ extended his 

 former investigations by taking into consideration changes of volume. 

 The velocity-potential is obtained in an infinite series in which the terms 

 are multiple integrals, which are integrable when the spheres move along 

 the line of centres. This was followed by a series of papers ^ in 1875 and 

 ]876, in which the consequences of the motion were developed more fully. 

 If inverse powers of the distance above the fourth be neglected, the vibra- 

 tions of one sphere produce only oscillations of the other, if that other 

 does not also vibrate. Simultaneous vibrations of the same period produce 

 mean forces of the second, third, and fourth powers of the inverse dis- 

 tance ; thus, two spheres with concordant pulsations attract proportionally 

 to the inverse square of the distance, and repel according to the same law, 

 if they are in opposite phases. If a, b be the radii of the spheres then the 



force on (h) due to pulsations of (a) is 1^ j( a~h '-^\ the density of 



the fluid being unity. Two oscillating spheres behave in the opposite man- 

 ner to two magnetic poles. These actions have suggested to Herr Bjerknes 

 analogies with electricity and magnetism, which he has illustrated by a 

 series of beautiful and striking experiments.^ It must be remembered, 



■ 'Sur le mouvement simultane des corps splieriques variables dans un fluide 

 Ind^fini et incompressible,' Forh. Vidensli. Christiania, 1871. 



■ ' Forelobige Meddelelser om de Kriif ter der opstaa, naar kuglef ormige Legemer 

 idet de udfore Dilatations; og Kontraktions Svingninger bcviige sig i et inkom- 

 pressibelt Fluidum. — Mirh. Vidensk. p. 386, Christiania, 1875. 



For an abstract and description of this, see Koenigsberger's Rejjeriorium fur reine 

 und an§ewandte Math. i. p. 264. ' Ueber die Druck-Krafte, die dvirch gleichzeitige 

 mit Contractionen und Dilatationen verbundene Bewegnngen von mehreren kugel- 

 formigen, in einer incompressiblen Fliissigkeit befindlichen, Korpern entstehen.' — 

 Gm. Nach. 1876, p. 245. 



» For descriptions of these, see ' Versuche iiber die scheiubare Anziehung und 

 Abstossung zwischen Korpern welche sich in Wasser bewegen.' Von Schiotz und B j erknes, 

 Gott. Nauh. 1877, p. 291. 



' Hydroelectricite et hydromagnetisme, r^sultats analytiques,' C. E. 1879. 



' Do. r^sultats experimentaux,' lb. 1879. 



' Experiences hydro dynamiques avec des corps vibrants et imitation dans un sens 

 invarse des forces de I'electricite statique et du magnetisme,' C. R. 1879. 



' Ph^nom&nes dits hydroelectriques et hydromagnetiques, theorfemes fondamentaux 

 et teur verification exp6rimentale,' Scaiicgg Soc. Phys. Paris (1877), Jovr. Phi/s. 

 Jloroh, 1880. 



* Hydrodynamiske analogier til de statlsk clektriske og do magnetiske Kriif cer,' 

 Naturen (Christiania) 1880. Also Forh. Sh. N. (Stockolm). 



' Sur I'imitationpar voie hydrodynamique des actions electriques et magnetiques,' 

 a R. 1881. 



On the same or analogous questions see also Hydrodynamic analogies to electricity 

 and magnetism by G. Forbes, Nature, xxiv. p. 360 (1881). This is a description of 

 Bjerknes' experiments. 



* Phenomlnes hydrodynamiques inversement analogues 3, ceus de I'electricite et du 

 magnetisme.' — Comptes Rendus, par M. Bertin, Ann. de Chimie ct de Phys. (5) xxv. 

 p. 257, 1882. This is a systematic description of Herr Bjerknes' work. 



'Experiences hydrodynamiques, imitation par les courants liquidcs, des pheno- 

 mSnes electromagnetiques et d'induction.' Decharme, C. R. xciv. p. 440, and p. 527. 

 • Do. des actions des courants electriques les uns sur les autres,' p. 643. ' Do. des 

 anneaux de Nobili, obtenus avec les courants electriques,' p. 722 (1882). 



« Backlund : Om en siirskild art af rorelse i en obegriinsad, osammantrykbar viitske, 

 i hvilken sammantrykbare kroppar iiro utspridda,' Lund. Arsshr. xv. 



Experimental Researches into the Pro2)ert'>xs and Motions of Fluids, mitli Theoretical 

 Deductians therefrom by W. Ford Stanley. London, Spon, 1881. 



