72 REPORT— 1881. 



These equations are made linear on the supposition that the compres- 

 sibility is small, and so that if p = pj, (1 + s) and p =po + Pqu^s, 

 quantities of the order 1/a, and sQi are neglected. It follows easily that 

 every point of the surface progresses with a velocity a, so that different 

 positions of the surface at different times form a system of parallel surfaces. 

 Supposing the hydrodynamical equations to hold on both sides and up 

 to the surface of discontinuity, he then shows how to determine the 

 function S = Sj — So on which the solution for ii depends. In the latter 

 part of the paper reflection at a rigid wall is considered. Rankine ' has 

 also touched on the same theory. 



(d) General Theory of the Motion of Solids in Fluid. 



Already, in 1843, had Stokes^ proved that if a body symmetrical with 

 respect to two planes at right angles to each other moved in a fluid 

 parallel to their intersection under tbe action of no forces the resultant 

 pressure of the fluid on it would vanish, provided the body and fluid were 

 originally at rest — in other words, that there is no I'otational motion in the 

 fluid. If, on the contrary, its motion Averc accelerated, it would experience 

 a resultant pressure equal to n x acceleration x mass of fluid displaced 

 by the body, where n depends alone on the /or)??- of the body and not its size. 

 A special case of this was also proved in 1854 by Hoppe,^ apparently 

 without knowledge of Stokes' paper. The case considered was that of 

 a solid of revolution, the equation of whose meridian section could be 

 thrown into a particular form.'* The fact that in every body there are 

 three directions which 230ssess the same property was explicitly stated 

 and proved by Kirchhoff in his paper referred to below. 



The starting point for the investigation of the motion here considered 

 was given by the publication in 1867 of Thomson and Tait's ' Natural 

 Philosophy.' Here the authors applied the Lagrangian equations of 

 motion of a dynamical system directly to the energy of the fluid, ex- 

 pressed as a quadi'atic function of the component velocities of the body, 

 which, referred to axes fixed in the body, has constant coefficients. By 

 this means the general properties of the motion of a solid of revolution 

 in a fluid, and moving in one plane, are deduced with the greatest ease. 

 In the general case there are twenty -one constant coefficients which may 

 be supposed known, and may either be found by analysis (theoretically 

 at least) or by experiment (just as coefficients of self- and mutual in- 

 duction in the case of electric currents) by finding the impulses necessary 

 to generate the unit velocities and their combinations two-and-two. The 

 general theory of the impulse as applied to fluid motion has been very 

 fully developed by Sir W. Thomson^ in his paper on vortex motion, before 

 referred to. 



We have already seen (p. 4) how he has shown that when the 

 motion can be produced from rest by application of forces to the solids 



' ' The thermodynamic theory of waves,' Trans. Hoy. Sue. 1870. 



" ' On some cases of Fluid Motion,' Trams. Camh. PMl. Soc, viii. p. 105 ; also in 

 Eeprint of Papers, p. .50. For a proof of this from the theory of stream lines and 

 many illustrations of other points in tiuid motion, see Froude in Nature, xiii. 



* ' Vom Widerstande der Flussigkeiten gegen die Bewegung fester Korper,' Pogg. 

 Ann., 93 (1854) ; also ' Determination of the motion of conoidal bodies through an 

 incompressible fluid,' Qva/rt. Jmir. Math.,\. p. 801. 



* See under special problems. 



' Trar)s R.S.E., xxv, 



