{(f)+^f'}» 



314 CELESTIAL MECHANICS. I. vil. 37- 



by the differential variation —ag^^, and Ihat SF:= (SF) — 

 ag^(p + a^V'; (SF) being the value of ^F which belongs to 

 the stratum in the state of equilibrium, and to the angles 

 6 -\-cai and nt + 'sr-^-av, and ^F' being the part of 2 F belong- 

 ing to the new forces, which act on the atmosphere in the 

 state of motion. 



Let f he~{^) + a^\ (f) being the density of the level stra- 



Ip 

 turn in the state of equilibrium. If we make — — y', we 



(f ) -^ 

 shall have -^= ^ \^ + agSy ; [since p zz Ig^ — Ig 



^ud^ JMKccg^M^ or, substituting (f) 



far f in the denominator, their difference being inconsider- 

 able in comparison with the whole quantity, =7^ — ^ -\- 

 agly\ Now in the state of equilibrium 



0=:iw2g|r + «5)sin (5 + aw) | ^ + (8 F)— ^!-^. Conse- 

 ^ -^ (f) 



quently the general equation of the motion of the atmos- 

 phere will become, in relation to the level strata, with re- 

 gard to which ^r is nearly evanescent, 



r^lL ( 2n sm cos 5 -— ) 



\ dt^ dt / 



\ ai^ dt r at f 



—g^(p—g^y'+ n^rsm-9 ^j «' — (/)>, a (/) being the varia- 

 tion of r corresponding, in the state of equilibrium, to the 

 variations au' and av' of the angles 9 and 'sr, [For ^V — 



^ becoming ={W)—ag^<p-\ra^r—^, and (^F)—^ be- 



ing = — Iw^S jir + as) sin (fi + aw) >2-_ ag^y' — 



an'r 



