1911] HYDRODYNAMICAL NOTES 21
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 force*. Ritzf has shown that the 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 x, y the general equation takes the form
(d?ldx2 + d2/dy" - 2 cos a d-jdx dy}" w = Q, ............ (19)
which may be differentiated any number of times with respect to x and y, with the conditions
=0, ............... (20)
iu = 0, dwfdx = 0, when » =0 ................ (21)
We may differentiate, as often as we please, (20) with respect to x and (21) with respect to y.
From these data it may be shown that ab the origin all differential coefficients of w with respect to x arid 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) $>w/dx'i = Q, d-w/dxdy = Q, and from (21) dhu/dy- = 0. Similarly for the third order from (20)
d3w/dccs = 0, d*wldotP dy = 03 and from (21)
dhu/dy3 = 0, d'wfdasdy* = 0.
For the fourth order (20) gives
d'w/dx4 = 0, d^w/ dx* dy = 0; and (21) gives
d*w/dy* = 0, d*w/dtJcdys = 0.
So far d4iv/dx2dy- might be finite, but (19) requires that it also vanish. This process may be continued. For the m + 1 coefficients of the wth order we obtain four equations from (20), (21) and m — 3 by differentiations of (19), so that all the differential coefficients of the mth order vanish. It follows that every differential coefficient of w with 'respect to x and y vanishes at the origin. I apprehend that the conclusion is valid for all angles « less than STT. 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 r, however high, would, I think, not have been expected without analytical proof.
* If indeed gravity act, iv=xzy^ is a very simple solution. f Ann. d. Phys. Bd. xxvin. p. 760, 1909. ...t memoir could be very simply derived from the expression for the