

1862.] 57 



gases, if calculated at temperatures not too close to the maximum 

 temperature of saturation. 



II. " On a New Method of Approximation applicable to Elliptic 

 and Ultra-elliptic Functions." Second Memoir. By CHARLES 

 W. MERRIFIELD, Esq. Communicated by WILLIAM SPOTTIS- 

 WOODE, Esq., F.ft.S. Received March 20, 1862. 



(Abstract.) 



Since my first memoir on this subject was read before the Society 

 in May 1860, Mr. Sylvester has published a method, more general 

 than mine, of applying rational approximation to facilitate the com- 

 putation of the integrals of irrational functions. His process, at 

 which he had arrived independently, included, ti majori, the one which 

 was the subject of my memoir. Aided by his papers, my subsequent 

 studies have enabled me to view the method with more generality, as 

 well as with more precision and completeness of detail, and I am 

 now able to present it in a sufficiently finished and practical shape 

 for the immediate use of the computer. I append auxiliary Tables 

 to facilitate its use in certain cases. 



I begin with the common radical form, starting from Mr. Sylves- 

 ter's definition of the approximants. Then decomposing the ap- 

 proximant into partial fractions by means of the roots of unity, and 

 increasing indefinitely the number of these fractions, I show that the 

 method is in reality the application of quadratures to a definite integral 

 which is substituted for the surd. The application of the process to 

 integration in like manner rests on the substitution, for the single 

 indefinite integral, of a double integral, definite in respect of one 

 variable, and indefinite for the other. The form of this double integral 

 is such that the indefinite integration can be performed directly ; and 

 the application of quadratures to the definite one is facilitated by a 



peculiar property of functions of the form , namely, that 



\-\-n sin 



the quadrature does not require the use of differences, but is obtained 

 simply from the mean of the ordinates. Legendre had previously 

 noticed and discussed this peculiarity, which is best illustrated by 

 effecting the quadrature by differential coefficients instead of differ- 

 ences. It will be found that these coefficients (which are all of odd 



