38 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 

 II. PERMANENCE OF THE VORTEX MOTION. 



We will next determine the variations of the velocities of rotation 

 g, ?j, C during the movement (of the surface) when only such forces are 

 effective as have a force potential. 



I note first in general that when ip is a function of #, y, z, t, and in- 

 creases by the quantity Sip, while the last four quantities increase by 

 S x, Sy, S z, and S t, respectively, we have : 



Sip=^St + ^Sx+^Sy+^Sz. 

 r dt n dx dy dZ 



If now the variation of ip during the short time tftfisto be determined 

 for one and the same particle of liquid, we must give the quantities 

 Sx, dy, 8z the same values that they would have for the moving parti- 

 cle of liquid, namely : 



S x — u St, dy = v St, Sz = w St, 



and obtain 



Sip dtp d'p , dtp , J<P 

 i^=T7 + u^- + v^- 4- te- 

 st dt T dx ^ dV dz 



I shall in the following always use the notation — ?- only in the sense 

 that -^-dt indicates the variation of rp during the element of time d t 



for the same particle of water whose coordinates at the beginning of 

 the time dt were x, y, and z. 



If by differentiation we eliminate the quantity p from the first of the 

 equations (1) and introduce the notation of equations (2) and substitute 

 for the forces X, Y, Z the expressions in equation (la), we obtain the 

 following three equations : 



^___ p 3u du ^du) 

 6t—*dx + 7] dy + ^dz 



#C z dw dw , )w 



or 



^ _ p du dv_ r dw ] 



St -^ dx +T? dx + ^ 



S? ? du dv r dw . ' 



M = ^ + 1? Ty + (: w\ ••••■ W 



St-^dz +r/ dz +Z ~dz~\ 



If 5, y, and C tor any particle of water are simultaneously zero then 

 also — 



St ~ St — St ~ °- 



Therefore those particles of water that do not already have a rotatory 

 motion will receive none in the subsequent time. 



