OF THE MOTIONS OF FLUIDS. 301 



considered being referred to the particles situated at the 

 surface of equilibrium, and not at the momentary surface, 

 on which the gravitation of the particles below it can have 

 no effect.] Then if we denote by a^V the part of SF which 

 relates to the new forces depending on the state of mo- 

 tion, whether they arise from the changes produced by the 

 motion, or from the attractions of the solid or the fluid, or 

 of any foreign body, we shall have, at the surface [of equi- 

 librium], S F=(S V)-ag^' + «S F. 



The variation ^n^^ \ir + as) sin (5 + aw) J is increased 



by the quantity an^^y. r sin «d, in virtue of the elevation of 

 the particle of water above the level of the sea ; [since Sr 

 becomes zz aJy', and 3^(r2 sin «d.)=2^r.r sin 25,]: but this 

 quantity may be neglected in comparison with — aghj\ be- 



cause even , the value of the centrifugal force, at 



9 



the equator, where it is greatest, is only a very small frac- 

 tion, equal to — -. Lastly, the variation of the radius r 



is so inconsiderable, for the different parts of the surface, 

 in comparison with its whole magnitude, that, for the pre- 

 sent purpose, we may make ^r-=zO\ and dividing the equa- 

 tion (jL) thus modified, by the coefficient a, we obtain the 

 equation of the proposition. 



374. Corollary 2. The equation of con- 





+ u 



tinuity will become 0=r' [ / + ^^^'( dd "^ 



cota 6 ) \ +^^^-^^ the density, after the time /, 

 being expressed by (p)+a?'- 



