420 
ME, C. W. MEEEIFIELD ON A NEW METHOD OE APPEOXIMATION 
quadratures to a definite integral, which we substitute for the surd proposed for 
evaluation. 
10. It would appear at first sight that a full apphcation of the method of quadratures 
in the ordinary way, with the help of difierences, would give better results than the 
mere summation of the ordinates. But this is not the case ; for the differences diverge 
immediately. If we use differential coefficients for the quadrature, instead of difierences, 
we have an opposite anomaly, namely that the correction of the summation appears to 
be absolutely nil, inasmuch as the differential coefficients which appear in the series are 
all of odd order, and the numerator of each of them contains the factor sin ® cos <p, which 
vanishes at both the limits 0 and Legendee has discussed this point. See the 
Appendix to the second volume of his ‘ Fonctions Elliptiques,’ p. 578. 
11. The application of the method to integrations, then, lies in the substitution for 
r M ^ F'fi 2Mr.dk.dt 
JJ„r^+(N-r^)cos^X.’ 
in which, since A and t are perfectly independent of each other, we may change the 
order of integration, thus obtaining 
2Mr . dt 
+ (N — r^) cos® Att 
dX ; 
and the rest of the operation depends upon our being able to perform the integration 
in } generally, and then to determine the integral in A by quadratures. The great 
advantage of the method turns upon the easy application of the method of quadratures, 
in consequence of our not requiring to difference the ordinates. 
12. One way of exhibiting generally the degree of convergence is as follows: 
N** always lies between 
Jr+ v/N)^+(r-^N)^ + A/N)^-(r- 
{r+ ^/N)‘— (r- ^N)t (r+ ^N)‘+ (r- -v/N)'’ 
and the error of either is therefore always less than their difference, 
]N^±4 
(r+ \/N)®'— (r— v^N)®* 
13. There is another mode, by which, in any given case, we may see how far it is 
necessary to carry our work in order to obtain a given number of decimals correctly in 
the result. Let be determined by the equation 
p- d^m __ P- 
Jo cos 5 ^~^J„ COS 6,’ 
/N r 
and let sin ~ or whichever may be less than unity ; then the mih. approximant 
/N 
will be . This is easily seen from the general term of the approximant, since 
