for Differential Equations of Mathematical Physics. 601 



§ 5. The parallelism with the flat plate demands that it" 

 the solution is to be obtained by an experiment of this 

 nature, the lateral loading will vary in magnitude and 

 in sign at different parts of the plate, a condition difficult 

 to realize in practice. By a process of graphical integration, 

 however, the problem maybe reduced to that of an unloaded 

 plate with modified boundary conditions. 



Consider, for example, the equation 



V 4 Z = /(*, y) ; 

 then a particular integral is 



Zi = ^dx 2 chj 2 \\og r 2 JJ log r 1 f(<c 1 p l )dat: l jy i ']. 



Tins equation may be evaluated graphical!}-, or, if convenient, 

 analytically, and by tire ordinary transformation the equations 

 whose solution is required can be reduced to 



V 4 Z = 0. 



This reduces the problem once more to an unloaded plate, 

 the method of fixing at the boundaries being determined 

 quite simply from the original conditions and the derivatives 

 of Z t . 



§ 6. Without attempting to enter into too great detail or 

 to evaluate the results in a given case, the nature of the 

 disposition of the stream-lines can be seen on general grounds 

 in any particular problem. Take, as illustration, the problem 

 of the motion of a cylinder about its direction of motion 

 moving at uniform speed do«n the centre line of a channel. 

 To determine the first term in the expansion outlined above, 

 we must consider an unloaded flat plate bounded externally 

 by the parallel walls of the channel and internally by the 

 section of the cylinder. Along the walls the velocity of 

 the fluid is zero, and consequently the plate must be clamped 

 horizontally along these boundaries. Round the inner 

 boundary the slope at each point with respect to w must 

 be zero, where x is along the channel, while with respect to y 

 it is unity. At the same time there must be no total reaction 

 on each boundary in consequence of the integral condition. 

 This implies that the cylinder must be. placed under the 

 influence of a pure couple about the axis of x of such 

 magnitude as to produce unit slope. Any slope, provided 

 it is small, may be taken as standard. This does not imply 

 that the velocity of the body is small, but ratlin- that the 

 analogy of the flat plate will only be valid when the deflexions 

 of the latter are small. The stream-lines corresponding with 

 the first term in the expansion can now be found by plotting 



