474 



The following communications were read : 



I. "On a new Method of Approximation applicable to Elliptic 

 and Ultra-elliptic Functions." By C. W. MERRIFIELD, Esq. 

 Communicated by the Rev. H. MOSELEY, F.R.S. Received 

 March 26, 1860. 



(Abstract.) 



I found my method on the known principle, that the geometric 

 mean between two quantities is also a geometric mean between the 

 arithmetic and harmonic means of those quantities. 



We may therefore approximate to the geometric mean of two quan- 

 tities in this way : Take their arithmetic and harmonic means ; then 

 take the arithmetic and harmonic means of those means ; then of these 

 last means again, and so on, as far as we please. If the ratio of the 

 original quantities lies within the ratio of 1 : 2, the approximation 

 proceeds with extraordinary rapidity, so that, in obtaining a fraction 

 nearly equal to V 2 by this method, we obtain a result true to eleven 

 places of decimals at the fourth mean. I name this merely to show 

 the rate of approximation. The real application of the method is to 

 the integration of functions embracing a radical of the square root. 



Suppose we wish to approximate to the integral of a function of 

 the form X\/ Y. The function is a geometrical mean between X and 

 XY. If, therefore, we obtain arithmetic and harmonic means to X and 

 XY, and again to these means, and so on, it is clear that our function 

 X \/Y will always lie between each pair of means of the series, the 

 arithmetic mean being always in excess, and the harmonic always in 

 defect. I now observe, 



(1) That if the functions X and XY both increase or both decrease 

 regularly with the independent variable, the integral of their geometric 

 mean will always be intermediate to their integrals, and also to each 

 pair of the integrals of the derived means. 



(2) That the derived series of arithmetic and harmonic means 

 contain no radicals, and are therefore integrable by resolution into 

 partial fractions, and that their integrals involve only logarithms or 

 inverse tangents. 



The last remark indicates that the method has no useful applica- 

 tion to functions of a simpler class than elliptic functions. It applies, 

 however, to all elliptic and ultra -elliptic functions, and to transcend- 



