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LVIT. On the Numerical Integration of Differential Equations. 

 By H. T. H. PlAGGiO, M.A., D.Sc, University College, 

 Nottingham *. 



I. Introduction and Summary. 



MANY physical problems lead to differential equations 

 which cannot be integrated in finite terms by any of 

 the usual devices. In such cases some form of approxi- 

 mation must be used. The earliest method of approximating 

 was by an infinite series, but this is often very tedious when 

 accurate numerical values are required. Runge (Math. Ann. 

 vol. xlvi. 1895) has given a formula for calculating approxi- 

 mately the increment of y corresponding to a small increment 

 of ./■, when x and //are connected by the differential equation 



and it is given that y = l> when a=a. 



Other approximate formulae have been given by Heun and 

 Kutta (Zeitschrift f Math. 21. Physik, vols. xlv. and xlvi.). 



In applying any of these methods to actual examples, it is 

 important to know how far wrong the result may be. Runge 

 assumes that when two steps of his approximations come 

 fairly close together, the error in the final result will be of 

 about the same order of magnitude as one-third of their 

 difference. But he does not give any definite upper limit 

 for this error. 



The object of the present paper is to supply this omission. 

 Four simple formula? are found which give four numbers, 

 between the greatest and least of which the required incre- 

 ment of y must lie. A new approximate formula is derived 

 from these. 



As an illustration this new formula is applied to the 

 example given by Runge. The result is more accurate than 

 those given by the methods of Runge, Heun, or Kutta. 



II. Limits between which the value of a definite integral lies. 



Let ¥(x) be a function which, together with its first and 

 second differential coefficients, is continuous (and therefore 

 finite) between x = a, and x= a -f h. Let F" (x) be of constant 

 sign in the interval. In the figure this sign is taken as 



* Communicated by the Author. 



