316 CELESTIAL MECHANICS. I. vii. 37. 



face of equilibrium, corresponding to the angles Q + au 

 and 'zar-f aw, increased by the elevation of the particle of 

 air above this surface ; the part of as\ which depends on 

 the variation of the angles 9 and -zir, being of the order 



, it maybe neglected in the preceding value ofy, and 



if 



we may consequently suppose in this expression s':=z<p; and 



ifwe then make (p^y, we shall have — - n 0, since the 

 •^ dr 



value of (p is then the same, with regard to all the particles 



situated in the same radius: besides, y itself is obvi- 



Tin 

 ously of the same order as /, or as — . : we shall therefore 



9 



have, for the value of y\ 



, , /dw' , Av' , u cos 0\ ^1 • / , 



y :=! — I ( -7-. +-r- + — ■ — 7- • consequently, smce u and 



v' are the same, for all particles originally situated in the 

 same radius, y' must be the same for all these particles. It 

 follows also, from these considerations, that the quantities 



2nr^'S7, s'm-9 -— , and n^r sin^ 0^ < s'—{s') ?, may be ne- 

 glected in the preceding equation of the motion of the 

 atmosphere, which may then be fulfilled by supposing u' 

 and 1;' the same for all the particles of air originally situated 

 in the same radius; and that the supposition of the con- 

 tinuance of all these particles in the same radius, during 

 the oscillations of the fluid, is consistent with the equations 

 both of motion and of continuity. In this case, the oscil- 

 lations of the different level strata are the same, and may 

 be determined by means of these equations; 



1. (^' -2» sin cos S ^') +r«g^.( sin'fl ^ + 2» 



