PAPER BY PROF. HELMHOLTZ. 09 



the factor — and all the terms of equation (la) by the factor— . Of 



the constants q, r, n, two are determined through the equations (2) aud 

 (2a) by the nature of the fluid, but the third, », is arbitrary so far as 

 the conditions hitherto considered come into consideration. 

 If the fluid is incompressible, then € is to be considered as a constant 



and t lr ==0 ' aiul the above equations then suffice to determine the motion 

 in the interior. 

 If the fluid is compressible, we can put 



p=a?e-c (3) 



'P=A*E-C ( 3a) 



where c and G indicate constants to be added to the pressure and which 

 have no influence on the equation la. 



For gases c and G are to be put equal to zero if the motion occurs 

 under such circumstances that the temperature remains constant. For 

 rapid variations of density in gases without equalization of temperature 

 (namely uon-adiabatic motions), the equations (3) and (3a) would only 

 apply for the case of slight variations in density. 



The equation (3a) is only satisfied by the above-given values for P 

 and E when 



A 2 =d*ri l . 



By this condition therefore the third constant, n, is determined. The 

 quantities a and A in this latter equation are the velocities of sound 

 in the respective fluids. These quantities must change in the same 

 ratio as the other velocities. 



If the boundaries of the fluid are iu part iu finitely distant and in 

 part given by moving or quiet, perfectly wetted, rigid bodies, and the 

 coordinates aud componeut velocities of these limiting rigid bodies are 

 transferred from one case to the other in the same manuer as has just 

 been done for the particles of fluid, then will the boundary conditions 

 for U, V, Wbe fulfilled when they are fulfilled for u, v, w. In this 

 1 assume that on completely wetted bodies the superficial layer of fluid 

 is held perfectly adherent; that therefore the component velocities of 

 the surfaces of the rigid bodies and those of the adherent fluid are 

 equal. 



For imperfectly wetted solids it is as a rule assumed that there is a 

 relative motion of the superficial fluid layers with respect to the solid. 

 In this case the application of our principles would require that a cer- 

 tain ratio be assumed between the coefficients of sliding superficial 

 friction of the fluid on the respective rigid bodies, and the internal 

 friction (or viscosity) of the fluid. 



Similarly the boundary conditions at the free surfaces of a liquid over 

 which the surface pressure is constant, would be satisfied in case no 



