EECENT PROGRESS IN HYDRODYNAMICS. 59 



the theory developed above to tbe cases of two-dimensional motion, and 

 of symmetrical motion in planes tbroagli an axis. In tbe first case, the 

 a,, o, give — one the planes in which the motion takes place, the other the 

 function introduced by Earnshaw (stream function) ; in the second case 

 the planes through the axis and Stokes' stream function. It is interesting 

 to see how these functions appear, though introduced in such a different 

 manner from those of Earnshaw and Stokes. The two-dimensional 

 stream function was also employed by Meissel, in a short paper, 

 ' Ueber einen speciellen Fall des Ausflusses vomWasser in einer verticalen 

 Ebene,'' apparently without being acquainted with its previous use by 

 Earnshaw and Stokes. Rankine, also, has used it in his paper on plane 

 water lines,^ where it enters as expressing the flow aci'oss any line. 



In the paper above referred to, Clebsch had reduced the case for 

 steady motion to one of making the energy a minimum, but he had not 

 succeeded in obtaining an analogous result for non-steady motion. This 

 question he takes up in a paper, published in Crelle,^ ' Ueber die Integra- 

 tion der hydrodynamischen Gleichungen,' but attacks the problem in a quite 

 different but original way. He introduces three functions, (j), 4^, m, so 

 that udx + vdy + ludz ^ df + md-<p. These can represent any general 

 motion of which a fluid is capable ; and the velocities and vortex 

 components, at any point, are given by equations of the form 



(/0 , d\L I 

 it = -^ + m -f, &c., 

 dx dx 



21 = ^ ('"'■ ^ ), &c., 

 d (y. z) ' 



or, expressed in quaternion language, vortex = i V ^m y\p. As in the . 

 earlier paper, he considers first the general case of any number of variables, 

 but we need only state the result as applicable to hydrodynamics. If 

 V denote the force potential, j^) the pressure, and p the density, the 

 equations make 



dx dy dz dt 



I(--f) 



a maximum or minimum where 



V - ^ = ^ + m^^ + i (^,2 + ^2 + ,„.) 

 p dt at 



and u, V, iv are expressed as above.'' The integrals of the equations 

 11 = X, V = y, IV = i, are shown to be in = const., \p = const., and a third. 

 It is clear, from the equation vortex = ^ V (v>n. V^)? that the vortex 

 filaments are given by the intersections of the surfaces m = const., 

 i// = const., and its strength is i T ^m. Tyi^ sin 6, where is the angle 

 between the surfaces, or 



*v lUi +-dij\ -^dzDUl +--j}--«- 



Between these two last papers of Clebsch, appeared the well-known 

 and remarkable paper of Hehnholtz in the same journal'^ on the integrals 

 of the hydrodynamical equations which correspond to vortex motion. 

 This will presently be noticed more fully under vortex motion, but atten- 



' Po/fff. Ann., 95, 1855. - Transactions Eoyal Sac, 186i. ' Crclle, Bd. Ivi. 

 ■• This is reallj' the same as Hamilton's theory of least action. ^ CreUc, Iv. 



