PROFESSOR KELLAND ON THE THEORY OF WAVES. 505 



force results from the impulse of the fluid behind that under consideration. We 

 suppose, in fact, that it is the difference of the impulses on the two sides of the 

 solid PN. 



Now the expression for the pressure at any point in the mass is this,* 



Of this expression the first term is gg(z—y) , which is the pressure due to 

 the action of gravity.. 



The second term is - 9f(^^-r^ , a quantity which depends on the mo- 

 tion of the particles in the neighbourhood of the point. 



In the theory of resistances, it is assumed that the motion is such as to admit 



of our writing — dt for da; ; that is, it is supposed that the motion of a particle 



is such as that, when it comes to another point in space after any interval, it 

 shall move just in the same manner as those particles do which are at the present 

 moment at that point. In other words, the motion is conceived to be steady, 

 that it is always the same at any particular point of space ; so that the integra- 

 tion for one particle during its successive stages shall be identical with the inte- 

 gration for different particles at the same instant. 



Such an hypothesis as this is not only convenient, but appears to be abso- 

 lutely correct in the case before us, for this pressure depends only on the varia- 

 tion of velocity of the particles immediately about PMN at the instant under con- 

 sideration, and if one hypothesis as to the future movements of the particles gives 

 the difference of the pressures, 



in going from point to point, any other hypothesis ought to give the same, pro- 

 vided the variation of velocity be the only thing which affects the pressure. 

 We may then assume as the value of ^ ; 



P being some function depending on % but not requisite to our calculation. 

 Similarly, p' =g Q («' -y)-^? «*" + P' 



=^K'^^>~^) "^K'^'"'^"^^*) ^^' 



* PoissoN, art. 647; Moseley, art. 203; Pratt, art. 562; Webster, art. 117. The general 



equation is, 



dp ( d^x\ ^ ( d-'y\ ^ ( d2:\ . 



VOL. XIV. PART II. • 4 P 



