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



If X, y be the rectangular co-ordinates of the axis of the cylinder, the 

 equations (vi) are equivalent to 



Jn 



(m+m')y= K P + TJ" 



where X t Y are the components of the extraneous forces. To find the effect 

 of a constant force, we may put 



(ix). 



The solution then is 



X = a + acos(?i + e), \ 



' ........................... w, 



provided ?& = Kp/(in + in') .................................... (xi). 



This shews that the path is a trochoid, described with a mean velocity g' jn 

 perpendicular to x*. It is remarkable that the cylinder has on the whole 

 no progressive motion in the direction of the extraneous force. 



70. The formula (1) of Art. 67, as amended by the addition of 

 the term A log z, 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 l9 A 2 , ... 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 limited application. 



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



* Greenhill, I.e. 



t A very powerful method of transformation, applicable to cases where the 

 boundaries of the fluid consist of fixed plane walls, has however been deve- 

 loped by Schwarz (" Ueber einige Abbildungsaufgaben," Crelle, t. Ixx., Gesam- 

 melte Abhandlungen, Berlin, 1890, t. ii., p. 65), Christoffel (" Sul problema delle 

 temperature stazionarie e la rappreseiitazione di una data superficie," Annali di 

 Matematica, Serie n., t. i., p. 89), and Kirchhoff (" Zur Theorie des Conden- 

 sators," Berl. Monatsber., March 15, 1877; Ges. Abh., p. 101). Many of the 

 solutions which can be thus obtained are of great interest in the mathematically 

 cognate subjects of Electrostatics, Heat-Conduction, &c. See for example, J. J. 

 Thomson, Recent Researches in Electricity and Magnetism, Oxford, 1893, c. iii. 



