OF THE MOTIONS Of FLUIDS. 305 



\ihrmm,0=^n^Bi r-}-as) sin {9-j-au) I ^ + (g F)— ^^\ (3^F) 



and (Sjo) being the values of S^Fand ^p which belong, in the 

 state of equilibrium, to the quantities r-\-as, 5 + aw, and 

 'sr + av, and which, in the state of motion, we may suppose 

 to become SF= (SF) + aSF', and ^p-{^p) + a^p' ; and 

 [since the variations and forces in the three different di- 

 rections afford independent equations,] we have 



V p/zz— 2nr sin^fi— 7 : [the other parts of the 



dr 



equation remaining the same as in the case of equilibrium, 



and therefore balancing each other]. Now it appears from 



dv 

 the equation (M) (375), that w ■-- is of the same order with 



yti 

 y or with s, and consequently with — ; the value of the first 



member of the present equation must therefore be of the 

 same order; and if we multiply this value by dr, and 

 find the fluent for the whole depth of the sea, we shall hav** 



for V — ^ a very small function, of the order — , besides a 



f r 



function of 5, -ar, and t independent of r, which we may call 



x\ consequently if in the equation (L) we only consider the 

 two variable quantities 6 and -ar, it will afford us the equa- 

 tion {M), with this difference only, that the second member 

 will become ^x. But since x is independent of the depth 

 of the particle, this equation becfomes equally applicable to 

 the surface and its neighbourhood, and the equations (M) 

 and (L) must in this case coincide with each other : hence 



we have hx—'^V — ghj, and consequently Z(V' — -JzzS^F 



— g^y; the SF' in the second member of the equation re- 



