366 Lord Kayleigh on the 



neglected. Thus, writing as usual 



u = dty/dy, v=— dyjr/dx, 

 we get from (34) 



• . d\. (Lj\. / * ^\ 



V ^=^~liy (42) 



Forces derivable from a potential do not disturb the 

 equation V 4 ^ — 0. In the analogy with a thin elastic plate, 

 already referred to, a place where dY/dx — dX/dy assumes a 

 finite value in the fluid problem corresponds to a place where 

 transverse force acts upon the plate. 



The simplest example of the finiteness of the second 

 member of (42) occurs when it is sensible at one point only. 

 This is the case of forces derivable from a potential 0, where 

 6 denotes the angle measured round the point in question. 

 It is to be observed that in the fluid problem the forces 

 themselves are not limited to the one point, but they have no 

 " circulation " except round that point. In the elastic problem, 

 on the other hand, the transverse force is limited to the one 

 point. 



The circumstance last mentioned renders the elastic problem 

 the easier of the two to deal with in thought and expression, 

 and we will accordingly avail ourselves of the analogy in the 

 investigation which follows. It is proposed to examine the 

 infinitely slow motion of fluid within an enclosure, which is 

 maintained by forces having circulation at one point only, 

 with the view of determining whether a contrary flow, or 

 backwater, is possible. In the analogous elastic problem we 

 have to consider a plate, subject at the boundary to the con- 

 ditions that w (the transverse displacement) and dw/dn shall 

 everywhere vanish, and disturbed from its original plane 

 condition by a force acting transversely at a single point P. 

 For distinctness we may suppose that the plate is horizontal 

 and that the force at P acts downwards, in which direction 

 the displacements are reckoned positive. At the point P 

 itself the principle of energy shews that the displacement is 

 positive, and it might appear probable that the displacement 

 would be also positive at all other points of the plate. A 

 similar conclusion is readily proved to be true in the case of 

 a stretched membrane of any shape subjected to transverse 

 force at any point, and -.also in one dimension for a bar 

 resisting flexure by its stiffness. But a consideration of 

 particular cases suffices to show that the theorem cannot be 

 generally true in the present case. 



For suppose that the plate fig. (4) is almost divided into 



