446 Mr. C. W. Merrifield on Approximation to the Integrals 



11. I trust that physicists will admit that we are justified in 

 concluding, from the foregoing facts, that in the inductive s}mrk 

 the point of light is to be attributed to the recomposition of the 

 electric charges accumulated in the extremities of the secondary 

 wire, while the luminous atmosphere is produced by the electric 

 fluids contained in the parts of the wire nearer to its middle point. 



Leyden, October 31, 1860. 



LXI. On Approximation to the Integrals of Irrational Functions 

 by means of Rational Substitutions. By Charles W. Merri- 

 pield, Esq.* 



IN his " Meditation on Poncelet's Theorem '* in the October 

 Number of this Journal, Mr. Sylvester has done me the 

 honour to mention a theorem of mine upon the subject expressed 

 at the head of this paper. I have now to communicate an ex- 

 tension of that theorem. It is but right to state that this exten- 

 sion has been suggested to me by Mr. Sylvester's remarks on 

 the connexion between his method and the Newtonian system of 

 approximation to the roots of equations. 



We had both taken the pure quadratic as our base of opera- 

 tions ; but while he took the method of continued fractions to 

 obtain his converging terms, I took (under another shape) the 

 method of successive substitution. I am compelled to differ 

 from him as to the practical advantages of the two methods for 

 the purposes of the computer. 



In my paper, which Canon Moseley communicated to the 

 Koyal Society this year, I started from the principle that the 

 geometric mean between two quantities is also a geometric mean 

 between their arithmetic and harmonic means, and again between 

 the arithmetic and harmonic means of those means, and so on, — 

 the series of arithmetic means on the one side, and the series of 

 harmonic means on the other, giving a continual approximation 

 to the geometric mean. Now this process is but a particular 

 case of the Newtonian approximation to the roots of an equation : 

 viz., let a be a first approximate solution, obtained by trial, of the 

 equation fx=0, and c&Wfx the differential coefficient offx; then 



a second approximation is a—i—^b; a third approximation will 



fb J a 



evidently be b— jj-r = c, and so forth. If we apply this method 



to the pure equation x n =p> the convergent terms which we obtain 

 are as follows : — 



* Communicated by the Author. 



