38 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



II. PERMANENCE OF THE YORTEX MOTION. 



We will next determine the variations of the velocities of rotation 

 ^, 7;, 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 //• is a function of j:, y, 5, #, and in- 

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

 6 x^ d y, 6 z, aud 6 t, respectively, we have : 



6>l-=^Ut-^^6x+^Sy+fsz. 

 ^t ,\r ?y <.^~ 



If now the variation of f during the short time (^iisto be determined 

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

 6.V, Sy, dz the same values that they would have for the moving parti- 

 cle of liquid, namely : 



6x = u dt^ Sy = V St, 6z = iv St, 



and obtain ; 



dt (It ' ;)jp ^y ^z 



I shall in the following always use the notation -/ onlv in the sense 



St 



that ~dt iudicat es the variation of ?/• during the element of time d t 



St ^ 



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

 the time cH were .r, y, and z. 



If by differentiation we eliminate the quantity j; 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 : 



S$ ^c)M c>u ^t'^w} 



^Sl = ^Jv+'^':n/-^^^ 



St] ^ ^v ?iv ,1y ,ov 



W=^k+'^¥+^^i^^ ^^^ 



St -^ dx'^ '^?y "^ - ci* 3 

 or 



St ~^ dx '^^^dx '^^dx I 



Sr] du dv ^?ic [ .0 . 



^t = ^-^ + '^^j + -^\ (^^'^ 



- st^^Tz^'^^-^^l^S 



If ^, ?/, aud C for any particle of water are simultaneously zero then 

 also — 



S^_S7j_ S^_ 

 St~St~ St ~ 



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

 motion will receive none in the subsequent time. 



