512 XI. HYDRODYNAMICS. 



tion (p and the curl of a vector function Q, whose components are 

 U, V, W. Accordingly let us put 



_ 



- 



64 ) 



dt dx dy 

 Finding first the divergence of q we have 



f>*\ j- du . dv . dw 

 bo) div. q % ho "~ -Q = 



the divergence of the curl part vanishing. But by 128, 5) we 

 know that if cp and its first derivatives are everywhere finite and 

 continuous, we have 



ST. a 7 



dr. 



Since q is continuous by hypothesis, div. q is finite. Consequently the 

 lamellar part of q is determined by its divergence. 

 Secondly finding the curl of q, 



dy 

 All 4- dU 4- dV 4- dW 



-^ U + + + 



1Z_ zu\ _ <Lfiv_ * <w_ 3 W \ 



Vx dy) dz\dy + dz dx) 



Since the vector Q is as yet undetermined except as to its partial 

 derivatives by equations 64), let us assume that it is solenoidal, or 



68) |f+| + 

 then we have 



69) 2^ = 



n = ~ 



the integrals of which are as above, 



J7=^ 



kft* 



