BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 321 



Next, to find the upper limit, it is always possible to proceed as follows : Make a 

 gness at A B P n , that is to say, write down as sharply oscillatory a set of numbers as 

 possible for the body values, such for example as Table V. 

 when extended to fill the region, and add to it such boundary IABLE V. 



values as will satisfy the boundary equations. Call this guess 

 Xi. Then xi = -A- n P n + A B _iP n _! + , &c. Now operate on Xi 

 with 2)' many times in succession, and after each operation 

 readjust the boundary values so as to satisfy the boundary 

 equations. By this process the coefficient of P n is increased j , , 



relatively to the coefficients of the other (P)'s in the 



expansion of the resulting table, because X n 2 is the greatest of the X 3 . So that the 

 table approaches a multiple of P n . From P n it is easy to find X n 2 . In any case a 

 rough approximation to X n 2 suffices. 



When the boundary values /3 t . . . /3 S vanish this labour is unnecessary, for then X n " 

 cannot exceed (see Appendix) the greatest value of X 2 pertaining to an integral of 

 ( + 5)' X 2 ) (f> = with the given size of co-ordinate differences, and with (f> vanishing at 

 infinity ; and this value of X 2 depends only on the form of 1)', and may be calculated 

 once for all. It will be denoted by X L 2 . Thus when 2/ = E*/&c? +$?/%* considerations 

 of symmetry show that the most oscillatory integral is Table V. extended similarly 

 in all directions, and from this we find X L 2 = 4/('S,x) 2 +4/('S?/) 2 . 



Having thus found limits between which Xj 2 , X/, ..., X a 2 must lie, it remains to 

 choose the (a)'s so as to make w small for all value of X 2 in this range. In practice 

 this has been done by drawing the graph of w for arbitrary (a)'s and altering them 

 or adding new ones until the maxima and minima of the curve were all sufficiently 

 small. Figs. 1 and 2 are graphs of w, representing two approximation processes 

 requiring equal amounts of arithmetical labour. In fig. 1 the ()'s are distributed 

 over a wide range.* In fig. 2 all seven ()'s are made equal to X L 2 . The curves show 

 that these distributed ()'s reduce the amplitudes corresponding to a wide range 

 of X 2 to less than one-tenth of their original value. On the other hand the ()'s 

 concentrated at X L 2 reduce the amplitudes in the neighbourhood of X L 2 much more 

 perfectly, but leave the (P)'s of graver period less affected. The allowable type 

 of curve for o> will depend on what is to be done with the integral of 2/< = when 

 obtained. If its space-rates are required it is more important to abolish the modes 

 of vibration having the largest values of X 2 than it would be if volume integrals alone 

 were needed. 



When the ratio of \i a /\ L a is large, as in large tables, it is difficult to remove PI by 

 the processes indicated by figs. 1 and 2. For example if 2)' = W/ftx 2 + 'S 2 /S?/ 2 , 

 and if the boundary is a square of ten co-ordinate differences side, on which 

 <f> vanishes, then the (P)'s having the lowest values of X 2 will not differ greatly 



* The advantage of distributing the (a)'s fairly uniformly was pointed out to me by Prof. 

 A. E. H. LOVE. 



VOL. COX. A. 2 T 



