RECENT PKOGHESS IN nYDRODYNAMlCS. 81 



theory of tins kind it is easy to see that forces will exist whose appear- 

 ance on any theory of surface-friction in the equations of stress is here 

 objected to. There can, howcA'cr, be little doubt that for small motions 

 at least the ordinary equations give trustworthy results. Maxwell, in a 

 note published at the end of a paper by Rohrs ' has pointed out a way to 

 obtain limits of error in some particular cases by the consideration of 

 eiTor- forces. 



The question of steady motion in viscous fluids has been considered 

 by Craig,^ who has proved that if everywhere v^i\ V^*?, V^c ("T, v> c, being 

 components of rotation) be zero, or in other words if x^hb dx + ^'^v dy + 

 X'-w dz be a perfect difi'erential, the motion is steady. In this paper 

 several interesting transformations of the equations of motion, and of 

 the dissipation function are given. Oberbeck^ had before this shown that 

 when the motion is very small and steady, the vanishing of \'^i, \'n, V^c is 

 a necessary consequence. 



Helmholtz'' has given a method by which in certain cases the motions 

 of one fluid with given conditions can be directly deduced from that of 

 another fluid whose motion is geometrically similar. Denoting by u v %u 

 the velocities at time t, at the point (x.y.z) of a fluid whose density is p, 

 pressure p, and coefiicieut of viscosity ^j, and using dashed letters to 

 denote the same quantities for another fluid, he points out that when 

 there are no external forces the equations of motion are also satisfied by 

 lii=qfi, pi=r p, U]^=nu, &c., x^^qx/n, &c. , pi=n-rp + const., ti=^qt I n'^ 

 where q. r. n are three constants, two of which, q. r, are determined by the 

 nature of the fluids, the other, n, is arbitrary for an incompressible fluid, but 

 for a compressible fluid n must be the ratio of the A'elocity of sound in the 

 second fluid to that in the first. In incompressible viscous fluids, in which 

 bodies are immersed, n will be determined by the ratios of the coefficients 

 of slipping at their surfaces. If this coefiicient is zero, or if there is no 

 viscosity, we may take into consideration the action of gravity, but then 

 n^=q. The resistances to the motion of similar bodies in the fluids are 

 in the ratio q^r : 1, whilst the rates of work done in overcoming the.se 

 resistances are as 7iq'^r : 1. Many interesting results flow at once from 

 the foregoing considerations, e.g., the fact that in waves on the surface of 

 a heavy incompressible fluid, the velocity of propagation of similar waves 

 in similar vessels is always, whatever their form, proportional to the square 

 root of the wave-length ; this follows at once by putting q=n^ whence 

 Xi:=n-x, t^-=nt. Helmholtz applies these considerations to the relations 

 between ships and their horse-power, birds and their muscular power, 

 and works out with numerical details the relations between ships and 

 similarly shaped balloons, with reference to volume, horse-power, and 

 tonnage. For these results the reader is referred to the paper itself. 



' 'Spherical and Cylindrical Motion in Viscon.? Fluid," Proc. L. Math. Soc. v. 

 p. 12.5. 



* 'Viscous Fluids,' Jour. Franhland Inst., Oct. 1880, p. 217. 'On certain possible 

 cases of steady motion in a viscous fluid,' Amer. Jour. Math. iii. p. 269. A similar 

 statement is also given by Graetz {Zeits.f. Math. u. Phijs. sxiv. p. 230 : ' Einige Siitze 

 liber Wirbelbewegungen in reibenden Fliissigkeiten'), but with an evident error as to 

 non-possibility of production of vortices in such a motion by conservative forces. 



' ' Uebrr stationare Fliissigkeitsbewegungen mit Ceriicksichtigung der inneren 

 Reibung,' Borch. Ixxxi. p. 62. 



■* ' Ueber ein Theorem, geometrisch iihnliche Bewegungen Hiissiger Korper 

 betreffend, nebst Anwendung auf das Problem, Luftballons zu lenken,' Monatsher. 

 Berl. (1873) p. 501. 



1881. 



