318 MR. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



indicates that this last one can hardly be more than 1 in 1000 in error. The precise 

 analytical theory of the vibrations of a square plate with clamped edges does not 

 appear to be given anywhere, but we may obtain an upper limit to L 4 C L 4 by the 

 method in RAYLEIGH'S " Sound," 89. Assuming W = (a?-x*) 2 (b 2 -y 2 ) 2 * we have 



= (31-5000) (a- 4 + Zr 4 ) + (18-0000) (&)-', 



when a = b this becomes 129G'0/(2a) 4 , which is greater than 1283/(2a) 4 obtained by 

 finite differences. 



3 - 2. Successive Approximation to the Integrals. Having illustrated the use of 

 simultaneous integral equations, let us pass on to methods which have this property 

 in common : that starting from a table of numbers, correct at the boundary, but 

 otherwise merely as near as one can guess, one proceeds by definite methods to 

 modify this table and thereby to cause it to approach without limit towards the true 

 finite-difference integral. 



Conditions. The following methods of approximation have up to the present 

 been applied only to a limited class of equations satisfying the conditions given 

 below. 



Let f be an arbitrary function of position having n body values i/ 1; t/ 2 , ..., \jj n , and s 

 boundary values &, /3 2 , . . . , /8,. Let the differential equation to be solved be 2)< = 0, 

 where ^S) is a differential operator, together with such boundary conditions as 

 make the problem determinate. Let 35 be approximately represented by the finite 

 difference operator 5)', so that thfe body equations are 



, ..., % = ...... (1). 



Then in order that the following justification of the approximation method may apply, 

 it will be shown in the Appendix that 3)' and the boundary-equations must be linear ; 

 and the body- and boundary-equations must be the condition that a certain positive 

 homogeneous quadratic function V of i|/ 1} i// 2 , . .., i|> n is a complete minimum. Also 

 /, though otherwise arbitrary, is limited on the boundary to be the difference of two 



functions of position both of which satisfy the said boundary-conditions. Under 



d = d 

 these circumstances / can be expressed in the form"/= 2 A A P* where P* is an 



integral of ( + 5)' X/) <j> = 0, which satisfies the same boundary-conditions as/, 

 X/ being a positive constant, the sign before 1)' being the same as that of i/ A in 

 ty^k. The proof of this fact and of various other properties of the (P)'s will be 

 deferred to the Appendix. The (P)'s may be called the principal or normal modes 

 of vibration of the system. V is analogous to potential energy. Some of the 



* I am indebted to Prof. A. E. H. LOVE for pointing out this method and for giving me the numerical 

 result for a square. 



