8 THE EQUATIONS OF MOTION. [CHAP. I 



where p and p are any pair of corresponding values for the 

 element considered. The constant 7 is the ratio of the two 

 specific heats of the gas; for atmospheric air, and some other 

 gases, its value is T408. 



10. At the boundaries (if any) of the fluid, the equation of 

 continuity is replaced by a special surface-condition. Thus at a 

 fixed boundary, the velocity of the fluid perpendicular to the 

 surface must be zero, i.e. if I, m, n be the direction-cosines of the 

 normal, 



lu + mv + nw = (1). 



Again at a surface of discontinuity, i.e. a surface at which the 

 values of u, v, w change abruptly as we pass from one side to the 

 other, we must have 



I (U-L u 2 ) + m (V-L v 2 ) + n (w l w. 2 ) = (2), 



where the suffixes are used to distinguish the values on the two 

 sides. The same relation must hold at the common surface of a 

 fluid and a moving solid. 



The general surface-condition, of which these are particular 

 cases, is that if F(x, y, z, i) = Q be the equation of a bounding 

 surface, we must have at every point of it 



The velocity relative to the surface of a particle lying in it must 

 be wholly tangential (or zero), for otherwise we should have a 

 finite flow of fluid across it. It follows that the instantaneous 

 rate of variation of F for a surface-particle must be zero. 



A fuller proof, given by Lord Kelvin*, is as follows. To find the 

 rate of motion (v) of the surface F(x, y, z, t) = 0, normal to itself, 



we write 



F(x + lv$t, y + mv&t, 2 + nvSt, t + St) = Q, 



where I, m, n are the direction-cosines of the normal at (#, y, z\ 

 whence 



./,dF dF dF\ dF 



* I * -1- + 1 '+ * '-3- I + 0. 



V dx 



Since 



-j- - 



dy dz J dt 

 dF dF dF 



* (W. Thomson) "Notes on Hydrodynamics," Gamb. and Dub. Math. Journ. 

 Feb. 1848. Mathematical and Physical Papers, Cambridge, 1882..., t. i., p. 83. 



