84 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



This example is merely given as an instance of discontinuous 

 boundary-conditions. 



The values of (/&amp;gt;, -^ can be at once found, if required, by 

 integrating (30). 



85. A very general formula for the functions (f&amp;gt;, ty may be 

 obtained as follows. It may be shewn that if a function f(z) be 

 finite, continuous, and single-valued, and have its first derivative 

 finite, at all points of a space included between two concentric 

 circles about the origin, its value at any point of the space can be 

 expanded in the convergent form 



2 + ................. (31). 



If the above conditions be satisfied at all points within a circle 

 having the origin as centre, we retain only the ascending series ; 

 if at all points without such a circle, the descending series, with 

 the addition of the constant A , is sufficient. If the conditions be 

 fulfilled for all points of the plane xy without exception, f (z) can 

 be no other than a constant A Q . 



Putting f (z) $ -\-ity, introducing polar co-ordinates, and 

 writing the complex constants A ny B n , in the forms P n + iQ nt 

 2t n + iS n , respectively, we obtain 



&amp;lt;/&amp;gt; = P -f 2&quot; r n (P n cos nd-Q n sin nff) + 2&quot; r~ n (R n cos n6 + S n sin nO) 

 *jr= Q -f 2 r n (Q n cos nO + P n sin n&) + ^r~ n (S n cos nd - R n sinn0) 



............... (32). 



These formulae are convenient in treating problems where we 

 have the value of &amp;lt;/&amp;gt;, or of -=? , given over the circular boundaries. 



This value may be expanded for each boundary in a series of sines 

 and cosines of multiples of 6, by Fourier s theorem. The series 

 thus found must be equivalent to those obtained from (32), 

 whence, equating separately coefficients of sin nti and cos nO, we 

 obtain four systems of linear equations to determine 



86. Example 6. An infinitely long circular cylinder of radius 



