[ 417 ] 
XIX. On a New Method of Approximation applicable to Elliptic and Ultra-elliptic 
Functions. — Second Memoir*. By Chaeles W. Meeeifield. Commumicated by 
W. Spottiswoode, Esq., F.R.S. 
Eeceived March 20, — Eead April 3, 1862. 
Since my first memoir on this subject was read before the Society, Mr. Sylvestee has 
published a method, more general than mine, of applying rational approximation to 
facilitate the computation of the integrals of irrational functions. This method, at 
which he had arrived independently, included, a 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 sutficiently finished and practical form for 
the immediate use of the computer. I have also computed auxiliary Tables, to render 
its application easier in certain cases. 
Any rational formula, which gives approximately the value of a function to be inte- 
grated, may be integrated in lieu of it, and the result will in general be an approximate 
value of the integral sought. But for such a process to be of any practical utility, the 
convergence of the fonnula must be excessive, for the complexity of the integral forms is 
so great that the labour would be enormous, unless the terms were very few in number. 
In the discovery of formulae sufficiently convergent for the purpose, lies the success of 
the method. 
We are by no means restricted to functions under a square root, or even to pure 
radical forms at all. The principle applies with equal generality to functions which are 
given implicitly as roots of equations, and thus to a class of differential equations ; and 
Mr. Sylvestee has well remarked that these formulae not only afford facilities for com- 
putation, as by a method of quadratures, but also enable us to assign superior and infe- 
rior limits to an integral, without losing its generality of form. 
I shall begin with the approximation to the square root, giving it in its general form, 
and explaining its exact analytical signification. I shall then show its application to 
Elliptic Functions, and how, in the ordinary cases, certain simple reductions can be 
effected, which greatly lessen the labour of computation ; and I shall give these reduc- 
tions for the cases more commonly occurring, with some examples and working formulae. 
I shall then add a short account of the extension of the method. 
The paragraphs in the first two sections of this paper bear a consecutive number for 
convenience of reference. 
* For the First Memoir, see the Philosophical Transactions for 1860, p. 223. 
.tIDCCCLXII. 3 L 
