RECEXT rilOGRESS IN nYDUODVNAMICS. 57 



Report on Recent Progress in Ifydrody nan tics. — Part I. 



Bij W. M. Hicks, M.A. 



At the meeting of the Association at Swansea, last year, I was requested 

 to draw tip a report on the recent progress of hydi-odynamics. I have 

 interpreted this to extend back to the time of the last report to the 

 Association, in 1846, by Professor Stokes. Though the period is 

 somewhat extended, and necessitates a leport perhaps excessively long, 

 yet it seemed to me that this objection would be counterbalanced by the 

 advantage of the different reports to the Association forming a connected 

 series. The period separates itself naturally into three divisions, of about 

 ten years each : from 1846 to the publication of Helmholtz' paper on vortex 

 motion, in 1856 ; from then to the appearance of Thomson & Tait's 

 ' Natural Philosophy,' in 1866, which introduced a new general method 

 into the treatment of hydrodynamics, by the application of the 

 Lagrangian Equations of .Motion ; and from the last period up to the pre- 

 sent time. I propose to consider the progress under the two heads of 

 (I.) General Theory and (II.) Special Problems, though stress of other 

 engagements has prevented my completing the second part in time for 

 this meeting. I hope, however, to complete it before the next meeting 

 of the Association. 



Under each head I have striven to keep as much as possible the 

 historical order in which the subjects have been developed, though, for 

 the sake of continuity, it has sometimes been necessary to deviate slightly 

 from this rule. It has not been an easy matter, out of such a rich field 

 of investigation, to decide what to include and what to leave out ; but it 

 is hojDcd that the most important results obtained dui'ing the last forty 

 years will be found referred to, although the report does not pretend to 

 be in any sense a complete bibliography of hydrodynamics. Several 

 subjects, whose boundaries are not sharply defined, have been neglected 

 altogether, as they would almost require reports to themselves ; such are, 

 for instance, the theory of sound and tides. The experimental side, also, 

 has not been considered, except in a few cases, in the way of reference. 

 These restrictions have been rendered necessary to keep the length of the 

 report within reasonable bounds. 



I. General Theory. 



Under this head I propose to notice successively (a) treatment of 

 General Equations for a perfect fluid, (&) Vortex Motion, (c) Discontinuous 

 Motion, (cZ) Genei-al Motion of solids in fluid (e) Viscosity, (/) Waves in 

 liquids. 



(a) General Equations of motion of a perfect fluid. 



In 1856, Clebsch published a paper ' (it is signed ' August, 1854 ') 

 on the motion of an ellipsoid in fluid. The first part of the paper is 

 devoted to a transformation of the ordinary equations when there is a 

 velocity potential to moving axes, and thence to orthogonal curvilinear 

 co-ordinates. He expresses the sui-face conditions in terms of the same 

 co-ordinates, and shows that the velocity potential is a linear function of 

 the quantities which would now be called generalised velocities and 

 rotations. He then gives the pressures in the same generalised form, 



■ CreUe, 1856. 



