RECENT PROGRESS IN HYDRODYNAMICS. 39 



Report on Recent Progress in Hydrodynamics. 

 By W. IM. Hicks, M.A. 



Part II. Special Problems. 



This second part of tlie report will deal with matters of more purely 

 mathematical interest than the first, and will chiefly comprise the con- 

 sideration of those particular solutions of the equation ^-(j) = o, which 

 satisfy conditions given over the boundaries of various surfaces, and the 

 determination of the effective inertia of the surrounding fluid when 

 solids of diSerent forms move in it. The problems here considered may 

 be regarded in different lights, according as the investigator has 

 accustomed himself to think from a hydrodynamical, an electrical, or a 

 conduction of heat point of view. Consequently, it will be found that 

 several of the hydrodynamical solutions will bo found in papers with 

 electrical or other titles. In the following the motion of a perfect 

 fluid in (a) two and (6) three dimensions, and (c) of a viscous fluid, will 

 be taken in order, the latter from a mathematical standpoint, without 

 reference to the experimental researches which have been carried out by 

 a large number of investigators. But, before passing on, it may be well 

 here to add a few remarks in the way of correction or addition to the 

 first part of the Report published last year. 



Professor Larmor has drawn my attention to the fact that the theory 

 of the ignoration of co-ordinates, mentioned on page 60, is essentially 

 due to Ronth, who gave the complete theory in his Adams' prize essay 

 'On the Stability of Motion' (p. 60). The application of the theory to 

 fluid motion is due to Thomson and Tait. The statement on p. 65, 

 * that the circulation round any closed curve in the fluid is equal to twice 

 the surface integral of the resolved part of the vortices perpendicular to 

 the surface over any surface whose boundary is the curve,' is a theorem 

 due to Thomson, is not correct. Beltrami • states that Hankel gave the 

 theorem in 1861, in a paper ^ which I have not been able to obtain ; but 

 it seems to have been given by Stokes, in 1854, in a Smith's prize ^ paper 

 for that year. This would, therefore, appear to be the first publication. 



In the consideration of viscosity on p. 81 a notice of a paper by 

 Helmholtz ' ought to have been given, in which he proves two general 

 theorems. These are that, if squares and products of the velocity be 

 neglected, and if the fluid be not supposed to glide over the surfaces of 

 bodies immersed in it, then, (1) if the motion be steady, the currents in 

 the viscous fluids are so distributed that the loss of energy due to 

 viscosity is a minimum, on the supposition that the velocities along the 

 boundaries of the fluid are given ; and (2) a floating body is in equi- 

 librium in a viscous fluid flowing with slow steady motion, if the loss is 

 also a minimum, when the velocities of the fluid along the surface of the 



' Sui princijnifondamentaK delV idrodinamica razionalc. 



^ Zur allgcmeineii Theorio der Bmvegwig der Fliissiglieiten. Gott. 1861. Now 

 out of print. 



' .See Camh. JJnir. Calendar for 1854, p. 415. 



* ' Zur Theorie der stationilren StriJme in reibenden FKLssigkeiten,' Verh. 

 natrirhist, Vereim. Heidelberg. V. p. 1. (18GS.) Also Collected Works, i. p, 223, 



