430 ON VIBRATIONS AND DEFLEXIONS OF [411
Accordingly
and
rl cos w/9 e"nx = dw x ZT, , dec 4nr — ^ log (1 4- e~Zx — 2e~a; cos £ 1 + e~2a: - Se-^ cos (y — vi) D ; .........
1 +e Zx — 2e * cos (y 4- 77)
From the above it appears that
W = a? log (1 + e~Zx - 2e~x cos (y + ??)} = as log /i must satisfy V4PP= 0. This may readily be verified by means of
V2log/i = 0, and V2 TF = x V2 log A, + 2d log A,/d».
We have now to consider the sign of the logarithm in (36), or, as it may be written,
1. °
ex + e~x — 2 cos (y + 77)
Since the cosines are less than unity, both numerator and denominator are positive. Also the numerator is less than the denominator, for
cos (y — 77) — cos (y + 77) = 2 sin y sin r? = + ,
so that cos (y — 'n) > cos (y + tj}. The logarithm is therefore negative, and dw/dx has everywhere the opposite sign to that of Zn. If this be supposed positive, w on every line y~ const, increases as we pass inwards from x — oo where w — Q to a? = 0. Over the whole plate the displacement is positive, and this whatever the point of application (??) of the force. Obviously extension may be made to any distributed one-signed force.
It may be remarked that since the logarithm in (37) is unaltered by a reversal of x, (36) is applicable on the negative as well as on the positive side of x = 0. If y = r/, x = 0, the logarithm becomes infinite, but dw/dx is still zero in. virtue of the factor x.
I suppose that w cannot be expressed in finite terms by integration of (36), but there would be no difficulty in dealing arithmetically with particular cases by direct use of the series (28). If, for example, r/ = ^TT, so that the force is applied at the centre, we have to consider
Sw~3 sin \n-rr . sin ny . e~nx (1 4- nos), .................. (38)
and only odd values of n enter. Further, (38) is symmetrical on the two sides of y = \TT. Two special cases present themselves when x = 0 and when y = -|TT. In the former w is proportional to , , ,.
smSy-..., ................. .(39)
and in the latter to
e~* (1 + x) + i e~sx(l + 3x) + ~e-sx (l + 5a>)+ .......... (10)
August 2, 1916. • • • •• • •s been omitted from the denominator; with Z=oo the corrected result agrees with (7) when x = Q, if B = nA. W. F. S.] side of the rectangle is relatively long. It seems therefore desirable to inquire more closely into this question.