ON QUADRATURES AND INTERPOLATION. 349 



tbe summation is not identical with the complete elliptic integral, as may 

 easily be ascertained upon trial — and there are many other functions 

 which present the same peculiarity. The paradox seems the greater, in- 

 asmuch as the numerical coefficients of the differential forms are highly 

 convergent, seeing that when r is large 



%VT2'-^ =l:4.^ = l:40nearly. 



The explanation, however, is, that the numerical coefficients introduced 

 by the differentiation performed upon u increase without limit, so that 



(lr^-2r+i («« — ''o) becomes really go — oo, which is necessarily indeter- 

 minate, and may (and usually will) exceed the corresponding factor in 

 the previous term in some ratio which is a multiple of r*, and which 

 increases without limit as r increases. Then the series finally becomes 

 divergent, and the paradox is solved. 



In many such cases, and notably so in the rectification of the 

 quarter-ellipse, the subdivision of the base, that is to say, an increase in 

 the number of ordinates, gives exti-emely rapid convergence towards the 

 true value. Thus for 7.; = sin 45'', if we take three ordinates only, viz., 



U.2 = 0-8660254 

 ^^,3 = 0-3535534 



1-7195788 

 This, multiplied hj -^w, gives for the length of the quarter ellipse 



1-350284. The value taken from Legendre's table is 1-3506438820. 

 If we were to use the parabolic rule we should have 



Mj = 1- 



4m.2 = 3-4641016 



Us =■ 0-7071068 



3 I 5-1712U84 



1-7237361 



This, multiplied by x ''•, gives 1-35382, which is not so good a result as 



we got before. The anomaly here is the counterpart to the one already 

 mentioned. Its explanation is, that if the ordinates are differenced, the 

 differences diverge at once, and, therefore, the series of which Cotes' rules 

 are a mere transformation is divergent from the beginning, so that the 

 more terms of it are taken, the farther from the trtith is the result. In 

 other words, the higher rules are worse, instead of better, than the poly- 

 gonal rule. This is an instructive examjjle of the advantage of a sub- 

 divided interval over a rule of a higher order. 



