312 ME. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



W. F. SHEPPARD, " A Method for Extending the Accuracy of Mathematical Tables," ' Proc. Lond. 



Math. Soc.,' XXXI. 

 KARL HEUN, " Neue methode zur approximativen Integration der Differentialgleichungen einer 



unabhiingigen Verariderlichen," ' Zeitschrift Math. u. Phys.,' No. 45, 1900. 

 WILHELM KTJTTA, " Beitrag zur niiherungsweisen Integration totaler Differentialgleichen," ' Zeit- 



schrift Math. u. Phys.,' No. 46, 1901. 



Further RICHARD GANZ, in a paper " tlber die numerische Auflosung von partiellen Differential- 

 gleichungen," ' Zeitschrift Math. u. Phys.,' No. 48, 1903, has extended the methods of RUNGE, HEUH, and 

 KUTTA to partial equations of the type considered in this section. Those of the first order he turns into 



theform a< ,/ a<A 



^ = F (x, y, <#>, jF), 

 OK oy I 



and starting from a boundary, where <J> is a given function of y, he expands this function as a power series 

 of y and integrates step by step in the x direction. The results he gives are of remarkable accuracy. 

 It is less accurate, but simpler, to work entirely by arithmetic in the manner illustrated in 2 2 below. 



2'L* A simple Process and its Possibility. For ordinary equations the necessary 

 and sufficient condition is that, for an w th order equation, the integral and all its first 

 n\ differential coefficients should he given at the boundary. This is almost obvious 

 at first sight. The complications in the following arise entirely from having to attend 

 to the correct centering of the differences an important thing in practice. Let the 

 equation be 



Then if all the quantities of which y is a function are given at x = x , we can 

 calculate d n <f>/dx" at x . Now, representing differentials by simple central differences, 

 draw up a table in columns. The subscripts denote distance from x . 



x. 



(S f /,) u (8^) 



.TO + ~2h <j,., h 



Each difference is centred at values of x halfway between those for the difference 

 of next lower order. Then, beginning with S"<, each difference is added to the one of 

 next lower order and the sum written down in the column of the lower order one, a 

 step h after it. This process gives a table with a diagonal boundary. The next 

 value of S"( is found from the difference equation and the process is repeated. The 

 difference equation is satisfied at those values of x where the highest difference is 

 tabulated. These processes involve moving certain differences through a step of ^h, 

 at the start and when satisfying the difference equation. Except for 8" and 8"" 1 this 

 is done by (8 r <) l/2A = (S r <) + |7i, (S r+1 <) . Note that this is a central formula with 

 respect to the step ^h to +%h. For S"" 1 and 8" the motion of %h is accomplished by 



* Revised April 20, 1910 



