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XIII. (hi a new Method of Approximation aiJplicahle toElliptic and Ult7'a-elliptic Functions. 
By Chakles W. Merrifield. Commimicated hy the Bev. H. Moseley, F.R.S. 
Eeceived March 26, — Eead May 24j, 1860. 
The difficulty of finding approximate Tallies of elliptic functions of the third kind has 
led me to consider a general method of approximation, which I believe to be new, at 
least in its apphcation to the evaluation of integrals of irrational functions. 
It depends on the known principle that the geometric mean between two quantities 
is also a geometric mean between their arithmetic and harmonic means. If we take 
any two positive quantities, we may approximate to their geometric means as follows : — 
Take the arithmetic and harmonic means of the two quantities, then again take the 
arithmetic and harmonic means of those means, and so on: the successive means will 
approximate with great rapidity to the geometric mean. 
To judge of the convergence of the method, I give, in the first two columns of the 
following little Table, the arithmetic and harmonic means thus derived from the numbers 
1 and 2 (which is the most unfavourable case that need present itself). The third 
column contains the difierence of the first two, within which lies the error of either. 
3 
4 
1 
2 
3 
6 
17 
24 
1 
12 
17 
204 
577 
816 
1 
408 
577 
235416 
665857 
941664 
1 
470832 
665857 
313506783024' 
Either of the fourth pair would thus give the square root of 2, correct to eleven places 
of decimals. 
This method finds its application in evaluating the integral ^fx.\/<px.dx, where y.V 
and (px are rational functions of x which both increase or decrease regularly, and have 
no singular values, within the limits of integration. If we find successive means, as 
above described, say A, and H^, between fx and fx.<px, then it is clear that, since 
fx\/ <px always lies between and H^, so ^fx.\/ <px.dx must lie between ^A^.dx and 
jH, .dx. Now A, and are each the product offx and a rational function of <px, and 
are therefore themselves rational functions of x. They are therefore always integrable. 
It is not to be denied that the application of this method to the functions which call 
