8H THE MECHANICS OF THE EARTTl's ATMOSPHERE. 



change [i. e., without discontinuity]. The difference (iti — 7T 2 ) will there- 

 fore in general increase on one side of the surface for increasing dis- 

 tance dn from this surface, but decrease, that is to say, become nega- 



d{CD\ — GO-i) 



tive, on the other side; and thus on the side where - -^- is positive 



we must have 'y (tti — 7r 2 )>0 or positive for every other direction dh, 



in which one moves from any point of the surface towards the same 

 side as dn. 



If dh is drawn toward the other side of the surface for which 

 7^ — ^2=0, then will 



l) {7t x — 7r 2 )<0, or negative. 



If now the difference is positive on that side of the surface designa- 

 ted by the subscript index l,then in case there is an infinitely small 

 protrusion of the boundary surface toward this side, this protrusion will 

 be pressed back by the exterior and greater tt, ; similarly an infinitely 

 small protrusion toward the negative side will also be pushed back, 

 since there, on the other hand, tt x diminishes more rapidly in the interior 

 of such protrusion. Therefore in both these cases the equilibrium is 

 stable. On the other hand, the equilibrium is unstable when the dif- 

 ference {tt x — 7t 2 ) on the side of tt\ is negative. 



Xow we need not form the differential quotients for the direction dn. 

 It suffices to form them for dr or dp, and to merely determine whether 

 the positive dr or dp look toward the side whose index is 1 or that 

 whose index i.s 2. 



By forming these differential quotients from the equation (3/) there 

 results 



The differential quotient is positive vhen 0, > 0,. The partial dif- 

 ferentiation with respect to r while p remains unchanged, indicates a 

 progress in an ascending direction parallel to the earth's axis; that is 

 to say, in the direction of a line pointiug towards the celestial pole. 



The equilibrium i.s stable when the strata containing the greater quantity 

 of heat lie at higher elevations on the side towards the celestial poles. 



We now form the other differentia] quotients 



*>- >4(f-f)--KVI) ■ • • • ^ 



= „ ["i*.T«d_?y-'-'] (4/ , 



If in these equations 0, indicates the greater quantity of heat, then 

 the equilibrium is stable when everywhere along the boundary surface 

 we have 



co?—co£. oof—Goi 



< J <\ ~>P- ,L W. 



