228 Calculation of Chladni's Figures for a Square Plate. 

 Thus, if we set 



V ~3(l + /x) V ' {b) 



we have 



V'=i? 1 s +2 ?1?2 +f ?2 2 + 1 ^ (7) 



In like manner, if 



T = ^T ; , (8) 



T^fcf + Wift + &■(? + ft) (9) 



When we neglect q 2 and suppose that q x varies as cos pt, 

 these expressions give 



6qP _ 96yA' (1Q) 



if we introduce a as the length of the side of the square. 

 This is the value found in 'Theory of Sound/ § 228, equi- 

 valent to Ritz's first approximation. 



In proceeding to a second approximation we may omit the 

 factors already accounted for in (10). Expressions (7), (9) 

 are of the standard form if we take 



A-i, b = 2 , o-l + i^, 



L = i, M-J, N = | + _g; 



and Lagrange's equations are 



(B-p>M) qi + (G-p*X)q a = 0,j ' ' ' ■ } 



while the equation for p 2 is the quadratic 



p 4 (LN-M 2 )+p 2 (2MB~LC-NA)+AC-B 2 = 0. . (12) 



For the numerical calculations we will suppose, following 

 Ritz, that /* = '225, making C = 11'9226. Thus 



LN-M 2 = -13714, AC-B 2 = 7-9226, 

 2MB-LC-NA = -2x4-3498. 



The smaller root of the quadratic as calculated by the usual 

 formula is '9239, in place of the 1 of the first approximation ; 

 but the process is not arithmetically advantageous. If we 

 substitute this value in the first term of the quadratic, and 



