NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 233 



3, A 3 # n -2, A 3 n_i, and A 3 x n vary. For example, in Table II it is easy to see 

 that A 3 sin 75 is almost certainly -3. It follows from 10.20, 1, 2 that 



3- 



[ x n+ i = 



After the adopted value of A^^+i has been written in its column the successive 

 entries to the left can be written down by simple additions to the respec- 

 tive numbers on the line of t n . For example, it is found from Table II that 

 A 2 sin 75 = -72, Ai sin 75 = 262, sin 75 = 9659. This is, indeed, the correct 

 value of sin 75 to four places. 



Now having extrapolated approximate values of x n+ i and y n+l it remains to 

 compute / and g for x = x n+ i, y = y n +i, t = t n+ i. The next step is to pass curves 

 through the values of / and g i or t = /+i, t n , t n -i, .... and to compute the inte- 

 grals (2). This is the precise problem that was solved in 10.30, the only difference 

 being that in that section the integrand was designated by y. On applying 

 equation 10.30 (9) to the computation of the integrals (2), the latter give 



X n+ i = X n + k [/+! - l&ifn+i - - A,/^ - - A 3 / n+1 . . . ], 



1 - -kig n +i - ^ A 2 n+1 - 



2 12 24 



where 



( fn+l = /On+1, 

 \ gn+l = g&n+l, 



The right members of (4) are known and therefore # n+ i and y n+l are 

 determined. 



It will be recalled that/n+, and g n+ i were computed from extrapolated values 

 of x n +i and y n +i, and hence are subject to some error. They should now be re- 

 computed with the values of x n+ i and y n +i furnished by (4). Then more nearly 

 correct values of the entire right members of (4) are at hand and the values of 

 #+! and yn+i should be corrected if necessary. If the interval h is small it will 

 not generally be necessary to correct x n +\ and y n +\. But if they require correc- 

 tions, then new values of / n +i and g n+ i should be .computed. In practice it is 

 advisable to take the interval h so small that one correction to /+! and g n +i is 

 sufficient. 



After #n+i and y^i have been obtained, values of x and y at t n+ i can be found 

 in precisely the same manner, and the process can be continued to t = t n+s , /n+4, 

 .... If the higher differences become large and irregular it is advisable to 

 interpolate values at the mid-intervals of the last two steps and to continue with 

 an interval half as great. On the other hand, if the higher differences become 

 very small it is advisable to proceed with an interval twice as great as that used 

 in the earlier part of the computation. 



The foregoing, expressed in words, seems rather complicated. As a matter of 

 fact, it goes very simply in practice, as will be shown in section 10.9. 



