322 



MR. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



from cos (-fanirx) . cos (ronny), and for these X 2 = (m 2 +n 2 ) roo^- So that the 

 ratios of X, a , X/, &c., to X L 2 when set in order of size will run 0'0247, 0'0617, 0'0987, 

 0'1234, &c., up to nearly unity. Referring to fig. 1 it is seen that the seven 

 approximations would reduce the amplitudes of Pj, P 2 , &c., in the ratios + 0'48, 

 + 0'07, 0'065. 0'08, &c., the rest never exceeding yjy and averaging about - 4 V 

 Of course continued approximation would gradually reduce the amplitude of P 1( but 



+ 1-0 



1 



o 



o 



o 



3 



FIG. I. SfrfAt Appf?ox/M/iT/o/vs 



D/STf?/BUTD. 



01 -^T 0-a 0-3 0-4 0-5 0-6 0V 0-8 0-9 



L 



FIG. 2. SEI/EN APPROXIMATIONS 



H-O 



+0-5 



Curves illustrating the process of approximation. 



in cases like these it may be well to make a guess at the form of P,, to estimate 

 X^ as in Appendix, equation (33), to find an approximation to the amplitude A! in 

 $\ $u = SA A P A by the Fourier method (see Appendix, equations (29) and (22)), and 

 so to remove the greater part of the first term of the series before the approximations 

 are begun. This has been done in the problem of the dam, 4. 



We have so far supposed Xj 2 , X/, ..., X n 2 unknown. If any one X* 2 of these be known, 

 then making a = X/ will entirely remove P t from the series. This process may 

 sometimes be useful for removing the gravest modes of vibration P 1; P 2 , &c. 



Since the value of cu is independent of the order in which its factors are multiplied 

 together, it follows that the result of < t+1 of a series of operations of the type 

 ^m+i = fym + Q-m' 1 2)'<m depends on the initial guess fa and on the values of the ()'s, 

 but not on the order in which the (a)'s are taken. The application of this result 

 to practice is slightly limited because the number of significant figures retained 

 is necessarily limited. 



In carrying out an approximation-process with a set of (a)'s designed to make 

 w small, it is frequently only at the last stage that the predicted improvement 



