Numerical Integration of Differential Equations. 597 



positive, making the curve concave upwards. LP, MQ, NR 

 are parallel to the axis of ?/, M is the middle point of LN, 

 and SQT is the tangent at Q. OL = a, LN = /<. 



Fisr. 1. 



M 



N 



Then the area PLNR lies between that of the trapezium 

 SLNT and the sum of the areas of the trapezia PLMQ, 

 QMNR. 



r a +h 



That is, I F(x)dx lies between 



h¥(a + ih) = A say, 

 and iA{F(a)+2F(a + iA) + F(a + A)} = B say. 



In the figure F"(a?) is positive and A is the lower limit, 

 B the upper. If ¥"(x) were negative, A would be the 

 upper limit and B the lower. 



As an approximation to the value of the integral it is best 

 to take, not the arithmetic mean of A and B, but fB + ^A, 

 which is exact when PQR is an arc of a parabola with its 

 axis parallel to the axis of x. It is also exact for the more 

 general case when F{x) = a + bx + cx 2 + ex z , as is proved in 

 most treatises on the Calculus in their discussion of Simpson's 

 Rule. 



III. Extension of preceding residts to functions defined by 

 differential equations. 



Consider the function defined by 



j~ =/(^ y) ; y= h wn ®n x=a, 



where f(x, y) is subject to the following limitations in the 



