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



Since f = dzjdw = 1 e* cos ty ie* sin -v/r, 

 it appears that for large negative values of the velocity is 

 in the direction of ^-negative, and equal to unity, whilst for large 

 positive values it is zero. 



The above formulae therefore express the motion of a liquid 

 flowing into a canal bounded by two thin parallel walls from an 

 open space. At the ends of the walls we have (f> = 0, -^ = + TT, and 

 therefore f= 0, i.e. the velocity is infinite. The direction of the 

 flow will be reversed if we change the sign of w in (5). The forms 

 of the stream-lines, drawn, as in all similar cases in this chapter, 

 for equidistant values of i/r, are shewn in the figure*. 



67. A very general formula for the functions <, 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 this space can be 

 expanded in the form 



f(z) = A + A 1 z + A a ?+ ... +8^ + 8^-*+ (1). 



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 . 



Putting f(z) = (f> + i^ ) introducing polar co-ordinates, and 

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

 R n + iSn, respectively, we obtain 



<f> = P + 2"r n (P n cos nO - Q n sin nO) + ^r~ n (E n cos nd+ S n smn0)\ 

 ty = Q + 2"r n (Q n cos nO + P n sin nd) + ^r~ n (S n cos w0-JZ B sinw0)J 



(2). 



These formulae are convenient in treating problems where we 

 have the value of </>, or of dfy/dn, given over the circular boun- 

 daries. 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 (2); 

 whence, equating separately coefficients of sin nO and cos 116, we 

 obtain four systems of linear equations to determine P n , Q n , R n , S n . 



* This example was given by von Helmholtz, Berl. Monatsber., April 23, 1868 ; 

 Phil Hag., Nov. 1868; Ges. Abh., t. i., p. 154. 



