144 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



Supposing that U, V, $ are each proportional to e i(M + ** } , we may take 3t///3< = is\fi, 

 and therefore, if we put 2(os/\ = a- and introduce the abridged notation of the 

 previous section, we may write the above equations m the form 



The equation of continuity is 



where h denotes the depth, and the height of the surface-waves. On substituting 

 for U, V from (13) and performing the differentiations with regard to t, <f> this 

 becomes 



or 



This equation is equivalent to the equation (17) of Part T. A second equation for 

 the determination of the two functions i/, is obtained from the pressure-condition 

 at the free surface. On reference to 2 of Part I., this condition is seen to lead to 



* = v'-9t + v ......... (15), 



where v denotes the surface-value of the potential due to the harmonic inequalities, 

 and v the surface-value of the disturbing potential. 



In order to effect the integration of these equations, we introduce two auxiliary 

 functions ^, y 2 , connected with i/ by the relation 



* s ...... (16). 



On applying the operator (D CT/A) to the two members of this equation, we 

 obtain in virtue of (11) 



(D - cr/t)* = (1 - ^)( A + cr)^ + (s 2 - 



/* s )* 8 - - (17). 



Now the functions 9 lt ? have as yet been subjected only to the single condition 



