BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 351 



to Prof. A. E. H. LOVE for several vital improvements which were introduced on 

 re-writing the paper, especially for the suggestion of the theory in the Appendix ; 

 also to Prof. L ARMOR, Prof. SAMPSON, and Dr. R. T. GLAZEBROOK for much valuable 

 criticism. 



APPENDIX. Properties of the Principal Modes of Vibration* 



In the theory of the approximation process we have assumed that 



(i) An arbitrary function f can be expanded in the form f SA^P^, where P k is an 

 integral of 1/P ft = X/P, and satisfies the same boundary condition as f does. 



(ii) That the X 2 are real, all of one sign, and lie in a finite range. 



(iii) That a certain one-signed function of position I can be found such that 

 SIP A P( = 0, SIP/ = 1, where S denotes a summation over the body points. 



It will now first be shown that if the system of difference equations satisfies certain 

 conditions, then the simultaneous equations 



$>>, = <), $>> 3 =0, ...$> =0t ....... (1) 



are equivalent to 



where V is a one-signed homogeneous quadratic function of \jj l ...^t n . Next, it will be 

 shown that the desired properties (i), (ii), (iii) can be deduced from the existence of V. 



No attempt will be made to determine whether the properties (i) and (ii) can hold 

 under more general circumstances when V does not exist.J However, V exists for so 

 wide a range of physical problems that it is well worth considering. 



Conditions of Existence of V. As equations (l) are to be equivalent to equations (2) 

 it must be possible to find a set of numbers i lt i 2 , ... i n such that 



,$>>, = |f for 1= l...n ........ (3). 



ayi 



gay 



And then, since = ^ is independent of the order of the differentiations, 

 dy t <% 



W 



* The dynamical analogy which is the basis of this section was pointed out to me by Prof. A. E. H. 

 LOVE. It has been introduced by POCKELS in his book 'Uber die Gleichung, A% + /fc 2 = 0.' See also 

 RAYLEIGH, ' Sound,' vol. I, chap. IV. 



t The boundary values in equations (1) are supposed to have been expressed in terms of the body 

 values by the boundary condition. 



I As to the property (iii), note that by considering the coefficients of the body values in the sums 

 S [IPj^'Pt], S [IPjtS)'P;], it is easy to show that these sums are equal if, and only if, the reciprocal 

 relation (4) holds. And if they are equal, then A t 2 S [IPjP*] = A/S [IP*Pj], so that SIP;?* = when A/ is 

 not equal to A^ 2 . But the reciprocal relation (4) is necessary to the existence of V. 



