762 Dr. J. R. Wilton on the Solution of 



But, in general, the boundary conditions are not alone 

 sufficient to determine ^, and we have to resort to other 

 means in order to obtain the final form of the solution. 



In the examples that follow the endeavour has been to 

 give a consistent exposition of the mode of attack on the 

 problems of most frequent occurrence; hence the inclusion 

 of a number of well-known results. 



Hydrodynamical Problems. 



2. If in hydrodynamical steady motion under forces whose 

 potential is £l(x } y) the curve (1) is a free surface, we obtain 

 the stream function (Earnshaw's) i/r by means of the 

 boundary conditions ^ = 0, 



where is constant, together with V 2/ ^ = 0. The result is 

 *=i£(X's+Y'')»{C+2a(X,Y)}M,. 



The theory of plane progressive waves may be based on 

 this result, but the work is practically identical with that 

 of the well-known method due to Stokes. 



3. The motion of a cylinder of any form in perfect fluid 

 at rest at infinity may be obtained with equal ease. 



Let the cylinder be moving with a velocity whose com- 

 ponents parallel to the axes are U and V, and let it be 

 rotating with angular velocity w. Then we have V 2 ^ = 0, 

 and on the cylinder 



OS OS ds V OS *3*/' 



i. e. ^r=-Vy-—Yx — ^w(x 2 +y 2 ) 



when # = t. And thus 



f=i{F(0)-F(T)} + iU(Y 1 + Y 2 )-!V(X 1 + X 2 ) 



where the function F is to be determined from the fact that 

 ^r is nowhere infinite and vanishes at infinity. 



As an illustration we take the familiar case of the elliptic 

 cylinder, for which 



X = ccosh Xcostj, Y=c sinhXsin77. 



