214 WISOOJfSIN" ACADEMY SCIEKCES, ARTS, AISTD LETTERS. 



or contracted by other vortex rings exactly as deduced from 

 theory."* 



In this memoir it is also demonstrated that the loroduct of the ve- 

 locity of rotation into the cross section of a tort ex filament is constant 

 throughout the ivhole length of the filament. Moreover, that a 

 vortex-filament can never end within a fluid, but must either re- 

 turn ring shaped into itself within the fluid, or reach to the bound- 

 aries of the fluid. 



Precisely similar theorems had been announced by Sir Wm. 

 Thompson in a paper on the Mathematical Theor}'- of Electro- 

 Magnetism in ISttT.f Thompson, in this paper, designates the 

 strength of an electric current by CJ, and then says: '" In a con- 

 tin uou,-:; current this quantity is of course tlie same for everg secdon; 

 and, as it is impossible that a continuous stream of electricity can 

 emanate from one body, and be discharged into another, the current 

 must be re-entering., or every continuous current must form what 

 is called '' a closed circuit." It is found by experiment that what- 

 ever be the dimensions or material of the different parts of the 

 conductor along which the current flows, provided always the di- 

 mensions of the section be small compared with the distances 

 through which the electro-magnetic action is observed, the quan- 

 tity (j has the same value for all parts of it; and even in the 

 places where the electro-motive force operates, as has been shown 

 by Faraday, as in the liquid of any ordinary galvanic battery, or 

 in a conductor in motion m the neighborhood of a magnet, the 

 electro-magnetic effects are observable, and, probably to exactly the 

 same degree; so that it would probably be found that a galvanic 

 circuit, consisting of a battery of small cells, arranged in a circu- 

 lar arc, and a wire completing the circuit by joining the poles, 

 would produce the same electro-magnetic effects at all points sym- 

 metrically situated with reference to the circle, irrespectively of the 

 part of the circuit, whether the cells or the wire; provided always, 

 that the distances considered be great, compared with either the 

 dimensions of a section of the wire, or of any of the cells made by 

 planes perpendicular to the plane of the circle, through its center." 



* Professor Tait's book on " The Recent Advances in Physical Science " has two 

 figures, one showing how these vortex rings can be produced, and the other what 

 ha direjti xnof rotatioi an 1 ru m a ■, it will be in a rin^ ")acd f >ru} 1 . 



t See Thompson's '' Eeprim of Papers on Electro- Statics and Magnetism" — p. 

 409 et seq. 



JFor current strength Thompson uses the Greek letter gamma. 



