Chladni's Figures for a Square Plate. 227 



was farther elaborated in ' Theory of Sound/ vol. ii. Ap- 

 pendix A. I had supposed that this treatise abounded in 

 applications of the method in question, see §§ 88, 89, 90, 91, 

 182, 209, 210, 265 ; but perhaps the most explicit formula- 

 tion of it is in a more recent paper *, where it takes almost 

 exactly the shape employed by Ritz. From the title it will 

 be seen that I hardly expected the method to be so successful 

 as Ritz made it in the case of higher modes of vibration. 



Being upon the subject I will take the opportunity of 

 showing how the gravest mode of a square plate mav be 

 treated precisely upon the lines of the paper referred to. 

 The potential energy of bending per unit area has the 

 expression 



3(1 



^,[(W+.<.-,> (C&) -££}]. <i. 



in which q is Young's modulus, and 21i the thickness of the 

 plate (§ 214). Also for the kinetic energy per unit area we 

 have 



T = p/iu> 2 , (2) 



p being the volume-density. From the symmetries of the 

 case w must be au odd function of x and an odd function 

 of ?/, and it must also be symmetrical between a; and y. Thus 

 we may take 



w = q l xy + q 2 xy{x 2 + if) + q-, xy {x* + y 4 ) + & x Y + . . . . (3) 



In the actual calculation only the two first terms will be 

 employed. 



Expressions (1) and (2) are to be integrated over the 

 square ; but it will suffice to include only the first quadrant, 

 so that if we take the side of the square as equal to 2, the 

 limits for x and y are and 1. We find 



ff(V 2 "0 2 dxdy = 16? 2 2 , (4) 



* " On the Calculation of the Frequency of Vibration of a System in 

 its Gravest Mode, with an Example from Hydrodynamics," Phil. Mag. 

 yoI. xlvii. p. 556 (1899) ; Scientific Papers, iv. p. 407. 



Q2 



