EECENT PKOGEESS IN HTDRODf NAMICS. 41 



(1845), Qaiuckci (185G), Auerbach^ (1878), and ■ Chwolson ^ (1878), 

 from the electrical point of view. 



As sources and sinks may be regarded as the origin of all non-cyclic 

 motion, so may the vortex-filament bo looked upon as the basis for cyclic 

 motion. The cases of one or two vortices, discovered by Helmholtz, 

 have been already referred to.^ When more than two are present the 

 general treatment of their motions involves mathematical difficulties of 

 calculation, though the theory is quite straightforward. Particular cases 

 have been worked out with much detail by W. Grobli''' (1877), in a 

 paper which has much interest from the number of figures it contains 

 illustrating the paths in certain cases. He considers the cases (1) of 

 three vortices («) equal, but one of different sign from the other two, 

 (/3) equal and of the same sign, (y) two equal and opposite and double the 

 third ; also the conditions that they shall always lie at the angles of a 

 triangle (o) of constant size and form, (e) of constant form (4) with two 

 equal sides ; (2) of four equal vortices with a plane of symmetry — which 

 comes to the same thing as two equal vortices in an infinite fluid 

 bounded by a plane ; and (3) of 2n equal vortices with n planes of 

 symmetry, or one vortex in the fluid bounded by two planes inchned 

 at an angle win. In this last case each describes the Cotes' spiral, 

 r sin nd = const. It would lead us too far to describe more fully the 

 results arrived at, which, after all, are only particular cases of the 

 general problem. The last question has also been discussed by Green- 

 hill^ (1878), who shows that a vortex of strength m, in an angle w/n, 

 will describe its Cotes' spiral as if it were a particle under the attraction 

 of a force varying inversely as the cube of the distance from the angle, 

 and strength = ^ (v^ — 1) i'^''^- 



The theory of the fluid motions resulting when portions of planes are 

 held in a stream has been referred to in the last report,^ and it will be 

 sufficient here to give for reference the cases already solved. The case of 

 fluid flowing from an infinite space into a canal bounded by two parallel 

 planes is historically the most interesting, being the first example of dis- 

 continuous motion which yielded to the genius of Helmholtz.* The only 

 other solutions at present known are those discovered by Kirchhoff.^ 

 These reduce to special cases of the two general jn-oblems (1) where fluid 

 issues from between two straight lines drawn in any direction from two 

 points, and (2) where a straight line is opposed in a stream of fluid at any 

 angle. The solution of the equation of continuity for all the space outside 



■ ' Ueber die Verbreitung eiries clektrisclien Stromes iu Metallplatten,' Pof/ff. Ann. 

 xcvii. p. 382. He considers space bounded by two infinite lines at right angles. 



- ' Ueber die Verbreitung stationarer electrisclier Strome in leLtenden Fliichen,' 

 Wied. Ann. iii. p. 498. 



^ 'Ueber das Problem der Stromverzweigimg in einer ebenen Platte,' 

 Sclilbm. Z. xxiii, p. 47. 



^ Brit. Ass. HejK, 1881, p. 64. 



* ' Specielle Probleme iiber die Bewegung geradliniger paralleler Wirbelfaden,' 

 Inavff. Diss. Gott. pp. 8G ; also, VicHeljahrschHft der naturforschcndcn GescUsclmft 

 in Zurich, sxii. 



* ' Plane vortex motion,' Quart. Jour. xv. p. 10. 

 ' Brit. Ass. Rep. 1881, p. 69. 



^ 'Ueber discontinuirliche Fliissigkeitsbewegungen,' Monatsh. Altad. Berl. 1868, 

 p. 215. Phil. Mag. (4) xxxvi. p. 337 ; also reprinted in Helmholtz' Wissen. Ahhand, 

 Bd. i. p. 146. 



» ' Zur Theoric freier Fliissigkeitsstrahlen,' Crclle, Ixx. p, 289 ; and reprinted in 

 Ges. Werke, p. 416, and Vorlesungoi U. 3lath. Fhi/sik. Vorles. 21, 22. 



