320 ME. L. F. EICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



function of position. (See Appendix.) Squaring both sides of (7), multiplying by I 

 and making the summation, we have by a property of the (P)'s (see Appendix, 

 equations (31) and (32)) 



r/ \ 2 \ / \ 2 \ / \ 2 \T 2 



(1-^) l-^)x...x(l-^)| . . . (9). 



l/\ 2/ a/J 



Now it has been found that by a judicious choice of atj, a a , ..., a t , the quantity 



[/ X 2 \ / X 2 \ / X 2 \"1" 



1 -- - 1 - - x ... x 1 -- ) may be made small for all possible values of X, 3 . 

 !/ \ / V |/J 



(Thus fig. 1 shows this done for a set of seven (a)'s. This graph was arrived at by 

 trial.) The error E t+1 of < t+1 may therefore be made small in comparison with that 

 of </>!. This is possible because the values of X t 2 lie in a finite range ; corresponding 

 to the fact that there are only a finite number of terms in the series SAjP*. A similar 

 process will not work with the infinite series of sines, Bessel functions and other 

 infinitesimal integrals of ($) X 2 ) P = 0. In choosing a l} a 2 , ..., a diagram of the 

 kind shown in figs. 1 and 2 is a great help. In this we take for the abscissa a 

 variable X~ which takes in turn the values X^, X/, ..., X,, 2 , and as ordiuate we consider 



(10). 



The value of w at X 2 = X A 2 is the ratio of the amplitude- of the vibration P* in the final 

 approximation </> (+1 to its amplitude in the initial guess. The individual factors 

 (1 X 2 /a,.) in a) represent straight lines, all cutting the vertical axis at 01 = I and the 

 horizontal axis at the points X 2 = a,, a s , ..., a ( . By bringing any adjacent paif a r and 

 ,.+! closer together the values of w corresponding to the range of X 2 between a r and 

 , +1 are diminished, provided that the other (a)'s remain fixed. It follows that by 

 judiciously spacing the (a)'s along the horizontal axis and by taking a sufficient 

 number of such points (that is of approximations) the successive maxima and minima 

 of oi can be made all less, in absolute value, than any finite quantity e however small. 

 When this is done 8 (</><+! </> a ) 2 I being equal by (9) to 2A/w* 2 where <a k is the value of 

 a) at X 2 = X/ must be less than e 2 2A r a . That is to say, the ratio of the error of the last 

 approximation <j) t+l to that of the initial guess <, being {SI(^ t+1 -^ B ) 2 /SI (<i-< K ) 2 } 1/2 , 

 is less than e, and e can be made very small. 



A knowledge of Xj 3 , X/, ..., X B 2 is not necessary, but it is necessary to know the 

 limits within which they range, or limits enclosing these. For the lower limit we 

 require an estimate of Xj 2 , which in the dynamical application is the square of the 

 frequency of the gravest mode of vibration multiplied by a constant depending on 

 density and elasticity. It is usually sufficiently close to take some boundary such as 

 a rectangle or a sector of a circle, for which the frequency is known, and which fits in 

 a rough way the irregular boundary under consideration. 



