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MR. L. F. RICHARDSON : APPROXIMATE ARITHMETICAL SOLUTION 



which is more general than (2) of 3'2'1 by reason of the factor y. The boundary 

 values of (P*) m +i are filled in so as to satisfy the boundary equations. Now (P k ) m 

 can be imagined as expanded in the unknown series 



(P.)., = 2B,Pj (4). 



On the diagram of 01 and X 2 (see fig. 3) equation (3) means that the straight line 



Fig. 3. 



representing a single process of -approximation may now be drawn in any way instead 

 of having to pass through w = 1, X 2 = 0. 



By suitably choosing lines all of which pass through <a = I, X 2 = X* 2 , we can reduce 

 the amplitudes of every P in the series except that of P^, which is left unaltered. 

 See, for example, fig. 3, where X/ has been given the particular value |-X L 2 . To choose 

 these lines we must know X/ at any rate approximately. For the first step an 

 approximation to X/ is therefore calculated from (P*)i in the way described in the 

 Appendix equation (33). Denote it by (X^. For the succeeding steps (X/) 2 , &c., 

 are calculated similarly from (P A ) 2 , &c. By Appendix equation (33) the errors in k" 

 are reduced more rapidly than those of P. The success of the method will depend 

 on the original guess (P/t)i, when expanded as 2AjP 7 , being free from (P)'s having X 2 

 nearly equal to X/. 



3 '2 '4. Error in the Integral Due to Incomplete Approximation. A general guide 

 here is the approximation process itself. If, for example, this has been such as to 

 diminish the amplitudes of all the P's to less than yg- of their former values, and if, 

 for all that, <j> has not changed by nine times the permissible error, we may conclude 

 that the process has been carried far enough.* 



* April, 1910. This is probable but not certain. Thus if 99 sin x + 100 sin (3z) becomes 

 9 sin x + 10 sin (3x) the value at x = JTT does not change but it is not therefore zero, 



