8688.] INDIRECT METHOD OF OBTAINING SOLUTIONS. 87 



If A = P + iQ, the corresponding terms in &amp;lt;f&amp;gt;, ty are 

 P \ogr-Q0, P0+Q\ogr, 



respectively. The meaning of these terms will appear from 

 Example 2 above. 2?rP is the cyclic constant of -\Jr, i.e. (Art. 69) 

 the total flux across the inner (or outer) circle; and %7rQ, the 

 cyclic constant of &amp;lt;, is the circulation in any circuit embracing the 

 origin. 



The formula (31), as amended by the addition of the term 

 (38), may readily be generalized so as to apply to any case of 

 irrotational motion in a region with circular boundaries, one of 

 which encloses all the rest. In fact, corresponding to each internal 

 boundary we have a series of the form 



where c, =a + ib say, refers to the centre, and the coefficients 

 A, A^ A^ &c. are in general complex quantities. The difficulty 

 however of determining these coefficients so as to satisfy given 

 boundary conditions is now so great as to render this method of 

 very little utility. 



Indeed the determination of the irrotational motion of a liquid 

 subject to given boundary conditions is a problem whose exact 

 solution can be effected by direct processes in only a very few 

 cases. Most of the cases for which we know the solution have 

 been obtained by an inverse process ; viz. instead of trying to find 

 a solution of the equation (5a) or (7) satisfying given boundary 

 conditions, we take some known solution of the differential equa 

 tions and enquire what boundary conditions it can be made to 

 satisfy. In this way we may obtain some interesting results in 

 the following two important cases of the general problems in two 

 dimensions. 



88. Case I. The boundary of the fluid consists of a rigid 

 cylindrical surface which is in motion with velocity V in a 

 direction perpendicular to its length. 



Let us take as axis of x the direction of this velocity V, and 

 let ds be an element of the section of the surface by the plane xy. 



