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48                                            ON  THE  CALCULATION  OF                                        [35
necessity of the first two terms in the expression of an arbitrary functioi .It would have been better to have mentioned them explicitly; but I d not think any reader of my book could have been misled. In  168 th inclusion of all* particular solutions is postulated, and in  175 a referenc is made to zero values of the frequency.
For the gravest tone of a square plate the coordinate axes are nodal, an Kitz finds as the result of successive approximations
w = %?>i -f '0394 (uiV3 + vl u3) - -0040 its vs - -0034 (wji>B + UM) + -0011 (usv5 + u,v3) - -0019 w50B ;
in which u stands for u(x) and v for u(y). The leading term u^, or xy, : the same as that which I had used ( 228) as a rough approximation o which to found a calculation of pitch.
As has been said, the general method of approximation is very skilful] applied, but I am surprised that Ritz should have regarded the method itse as new. An integral involving an unknown arbitrary function is to be mad a minimum. The unknown function can be represented by a series of know functions with arbitrary coefficients  accurately if the series be continued t infinity, and approximately by a few terms. When the number of coefficient also called generalized coordinates, is finite, they are of course to be dete; mined by ordinary methods so as to make the integral a minimum. It \vt in this way that I found the correction for the open end of an organ-pipe-using a series with two terms to express the velocity at the mouth. Th calculation was further elaborated in Theory of Sound, "Vol. II. Appendix 1 I had supposed that this treatise abounded in applications of the method i question, see  88, 89, 90, 91, 182, 209, 210, 265 ; but perhaps the mo; explicit formulation of it is in a more recent paper J, where it takes almos exactly the shape employed by Kitz. From the title it will be seen th* I hardly expected the method to be so successful as Ritz made it in the cas of higher modes of vibration.
Being upon the subject I will take the opportunity of showing how tl gravest mode of a square plate may be treated precisely upon the lines of tl paper referred to. The potential energy of bending per unit area has tl expression
7 _    ^    r/s w + 9 n _ A [(.. d*w V _ *2 dh(J\~\
3(1 -/!,          W   + ^L                                   ~
* Italics in original.
f PiiiL Trans. Vol. CLXI. (1870) ; Scientific Papers, Vol. i. p. 57.
 "On the Calculation of the Frequency of Vibration of a System in its Gravest Mode, wi an Example from Hydrodynamics," Phil. Mag. Vol. XLVII. p. 556 (1899) ; Scientific Papers, Vol. 3 p. 407.orated screen, throws some light upon this question.