of the Motion of Fluids, ^-c. 411 



9. We have seen Art. 5. that (p is generally a function ot' 

 r and t, the form of which at a given instant is constant when 

 r is made to vary through a very small space; and that the 



velocity is -t^ • 



Hence if ^ = (r, #), (p'-<j) = <p {r, t) - cf) (r, t) 



Here r'-r may be considered the increment of a line s, drawn 

 from a fixed point, always in the direction of the motion of 

 the particles through which it passes. 



Hence d.-fr = -rr as; 

 at at 



Consequently, a- hyp. log. p = T -f-j-^ ds - ^ -/(«)• 



This equation is readily applicable, whenever the motion has 

 attained to such a state, that the velocity of every particle 

 passing through the same point, is the same in quantity and 



direction ; for on this supposition ^7 = at every point, and in 

 consequence, 



a' hyp. log. p= V-%-f{t) (K). 



in whicli a series of waves will act upon a small solid, such as a slender chord, is not 

 correctly stated in Art. 20, of that paper : it may be shewn, however, that the action !< 

 the same in kind as what is there stated, but dift'ercut in degree, and that the equation 

 for determining the vibrations of the chord is of the proper form. 



