8 THE EQUATIONS OF MOTION. [( HAP. I. 



where the suffixes are used to distinguish the two sides of the 

 surface. By subtraction we find 



I (X - w 8 ) + m (v, - v a ) + n (w, - w 8 ) = (13). 



The same relation holds at the common surface of two different 

 fluids in contact; and also, since in the proof of (11) no assumption 

 is made as to the nature of the medium of which (10) is a boundary, 

 at the common surface of a fluid and a moving solid. 



The truth of (13), of which (12) is a particular case, is other 

 wise obvious from the consideration that the velocity normal to 

 the surface must be, in each of the cases mentioned, the same on 

 both sides. 



11. The equation (11) expresses the condition that if the 

 motion be continuous the particles which at any instant lie in the 

 bounding surface lie in it always. For (11) expresses that no fluid 

 crosses the surface -Z^= 0; and the same thing necessarily holds of 

 every surface which moves so as to consist always of the same series 

 of particles. If then we draw a surface parallel and infinitely close 

 to F=0, and suppose it to move with the particles of which it is 

 composed, the stratum of fluid which is included between this and 

 *= 0, and which in virtue of the continuity of the motion remains 

 always infinitely thin, must always consist of the same matter ; 

 whence the truth of the above statement. 



It has been suggested that (11) would be satisfied if the part 

 icles of fluid were to move relatively to the surface F= in paths 

 touching it each at one point only. The above considerations shew 

 that this is not possible for a system of material particles moving 

 in a continuous manner ; although it would be so for mere geo 

 metrical points which might coincide with and pass through one 

 another*. It is, indeed, difficult to understand how, in the case 

 supposed, the particles which are receding from the surface are to 

 keep clear of those which are approaching it. 



12. In the above method of establishing the fundamental equa 

 tions we calculate the rate of change of the properties of a definite 



* The student may take as an illustration the motion of a series of points 



given by the formulae 



u .r, v c, 10=0, 



the upper sign in u being taken for points receding from the fixed boundary x = 0, 

 the lower for points approaching it. 



