226 Lord Rayleigh on the Calculation of 



tones of the plate may be expressed by the formulae 



w mn = u m (x)u n (y) -\-u m {y)u n (x), 



w'mn = u m {x)u n {y)—u m {y)u n {x), 



the functions u being those proper to a free bar vibrating 

 transversely. The coordinate axes are drawn through the 

 centre parallel to the sides of the square. The first function 

 of the series ?/ (^) is constant ; the second Ui(x) = w . const. ; 

 u 2 (#) is thus the fundamental vibration in the usual sense, 

 with two nodes, and so on. Ritz rather implies that I had 

 overlooked the necessity of the first two terms in the ex- 

 pression of an arbitrary function. It would have been better 

 to have mentioned them explicitly ; but I do not think any 

 reader of my book could have been misled. In § 168 the 

 inclusion of all * particular solutions is postulated, and in 

 § 175 a reference is made to zero values of the frequency. 



For the gravest tone of a square plate the coordinate axes 

 are nodal, and Ritz finds as the result of successive approxi- 

 mations 



to = u l v 1 -f- "0394 (u 1 1> 3 -f- v Y u s ) 



- -004:0 i/ 3 r 3 - -0034 (u, v h + i# 5 v,) 



+ -0011 (ti, r 5 + i/ 5 17 8 ) - '0019 u, v 5 ; 



in which u stands for n(js) and v for u(y). The leading- 

 term Mit?!, or ocy, is the same as that which I had used 

 (§228) as a rough approximation on which to found a calcu- 

 lation of pitch. 



As has been said, the general method of approximation is 

 very skillfully applied, but I am surprised that Ritz should 

 have regarded the method itself as new. An integral 

 involving an unknown arbitrary function is to be made a 

 minimum. The unknown function can be represented by a 

 series of known functions with arbitrary coefficients — accu- 

 rately if the series be continued to infinity, and approximately 

 by a few terms. When the number of coefficients, also 

 called generalized coordinates, is finite, they are of course to 

 be determined by ordinary methods so as to make the integral 

 a minimum. It was 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. The calculation 



* Italics in original. 



f Phil. Trans, vol. 161 (1870) ; Scientific Papers, i. p. 57. 



