Lord Riiyleigh : I]ylrodynamical Notes. 195 



solution is not developable in powers of x and y, and it may 

 be worth while to extend the proposition to the more general 

 case when the axes, still regarded as lines of clamping, are 

 inclined at any angle a. In terms of the now oblique 

 coordinates «r, y the general equation takes the form 



(&lda* 4- tP/dy* - 2 cos a d 2 /dx dy)*w = 0, , . (19) 



which may be differentiated any number of times with 

 respect to X and y, with the conditions 



w = Q, dw/dy = 0, when ?/ = 0, . . (20) 

 w?=±0, dw/dx=0, when#=0, . . (21) 



We may differentiate, ns often as we please, (20) with 

 with respect to a and (21) with respect to y. 



From these data it may be shown that at the origin all 

 differential coefficients of %o with respect to a and y vanish. 

 The evanescence of those of zero and first order is expressed 

 in (20), (21). As regards those of the second order we 

 have from (20) d 2 w/dx 2 ^0, d 2 w/dxdy = 0, and from (21) 

 d 2 w/dy' 2 = 0. Similarly for the third order from (20) 



d\cjdx* = 0, dhejdxhly = 0, 

 and from- (21) 



dhvjdy* = 0, dhv/dx dy 2 = 0. 



For the fourth order (20) gives 



d' w/dx 4 =e 0, dhvldxUy =k 0, 



and (21) gives 



dhv/di/ = 0, dhv/d.v dy 5 =e 0, 



So far d 4 w/dx 2 dy 2 might be finite, but (19) requires that it 

 also vanish. This process may be continued. For the m + 1 

 coefficients of the mth order We obtain four equations from 

 (20). (21) and rrt—'d by differentiations of (19), so that all 

 the differential coefficients of the ??zth order vanish. It 

 follows that every differential coefficient of W with respect 

 to x and y vanishes at the origin. I apprehend that the con- 

 clusion is valid for all angles ot less than 2v. That the 

 displacement at a distance r from the corner should diminish 

 rapidly with r is easily intelligible, but that it should 

 diminish more rapidly than any power of ?\ however high, 

 would, I think, not have been expected without analytical 

 proof. 



O 2 



