502 XL HYDRODYNAMICS. 



where P has the value of 177, 5). In the left-hand member occur 

 in this and the companion equations the expressions 



i /. _ *V > 



'2 \aa; ds// ~ S, 



which represent the components of a vector co, which by comparison 

 with the expressions for & x , & y , & z in 169, 77) is seen to be the 

 angular velocity of rotation of a particle or the vorticity. Let us 

 accordingly write, putting 



<Tr7\ 



27) 



Multiplying respectively by ^w, (>v, ^w; and adding, the terms in 

 1, 17, disappear and integrating over any volume, we have 



- / / [ cos (wa;) + cos (ny) + c cos ()] ( UQ p) dS 



^ -L ? ^") -L 



~ ' 



Introducing the value of U and of its multiplier in the last integrand, 



o 



-ST* 



29) 



o 



which by equation 10) is equal to - -ST* an ^ transposing, we obtain, 



If the applied forces are independent of the time, -^r = 0, and we 

 may write the left-hand member, 



