from which it Is again evident that if 



gP > U'V4 



everywhere in the fluid exclusive of the interfaces and the free surface 

 (if there is one), the flow must be stable, 



IV. EXTENSION OF HOWARD'S SEMI-CIRCLE THEOREM 

 Equation (9) can be written as 



(pW^F')' + p(pg - k^W^)F = 0. 



Multiplying this equation by F , the complex conjugate of F, inte- 

 grating throughout the fluid domain and using the boundary or inter- 

 facial conditions (11), (12), and (13), we have 



J^w'LlF'l'+kVl'] -ypgMF|'-^gA.,-^|F|' = 0, 



(20) 



in which the summation is over the surfaces of density discontinuity, 

 and the integrals extend throughout the fluid exclusive of the surface 

 of discontinuity in density. The real and imaginary parts of (20) are 



yp[(U-c,)' - Ci'][ iF'l' +k'|F|'] -^Pg^lFl^ -^gAjplFl" = 0, 



i 



lc.^'^{\3 - c,)[ |F'|^ +kVl^] =0- 



(21) 

 (22) 



Writing 



Q = p[|F'|'+k'|F|'], 



we obtain from (22) 



rUQ=c Co, (23) 



then from this and from (21) we obtain 



yU^Q= (c/ +C(2)yQ +ygpP|F|^ +^gAip|F|^ (24) 



i 

 If a and b are respectively the minimum and the maximum of U, 



224 



