194 Lord Rayleigh : Hydrodynamical Notes, 



For the D-solution we may take 



ty=r* sin 3 ^0. 



.... (15) 



Here *fy retains its value unaltered when 2tt — is substituted 

 for 0. When r is given, yfr increases continuously from 

 # = to 6 = 7r. On the line = ir the motion is entirely 

 transverse to it. This is an interesting example of the flow 

 of viscous fluid round a sharp corner. In the application to 

 an elastic plate yfr represents the displacement at any point 

 of the plate, supposed to be clamped along = 0, and other- 

 wise free from force within the region considered. The 

 following table exhibits corresponding values of r and 6 such 

 as to make i|r = 1 in (15) :- — 



0. 



r. 



e. 



r. 



180° 



100 



00° 



640 



150° 



123 



20° 



10* x 3-65 



120° 



2-37 



10° 



10 c x2-28 



90° 



8-00 



0° 



00 



When n = 2, (12) appears to have no significance. 

 When n = '3, the dependence on is the same as when 

 n — 1. Thus (14) and (15) may be generalized: 



yfr = (Ar* + Bri) cos \0 sin 2 \0, 

 ■f={AV£ + B'rf)sin 3 i0. 



(16) 

 (17) 



For example, we could satisfy either of the conditions i/r = 0, 

 or dyfr/dr = 0, on the circle r=l. 



For /i = 4 the D-solution becomes nugatory ; but for the 

 0- solution we have 



^ = Cr 2 (l- cos2<9) = 2Cr 2 sin 2 = 2C> 2 . 



(IS) 



The wall (or in the elastic plate problem the clamping) 

 along = is now without effect. 



It will be seen that along these lines nothing can be done 

 in the apparently simple problem of a horizontal plate 

 clamped along the rectangular axes of x and y, if it be 

 supposed free from forcer*. Ritz f has shown that the 



* If indeed gravity act, w=x 2 y 2 is a very simple solution, 

 t Ann. d. Phys. Ed. 28 ; p. 760 ; 1909. 



