[fnn Uw Pnemtutgt o/ the Cambridge Philosophical Society, Vol. in. 1877.] 



LXXXVI. OH Approximate Multiple Integration between Limits by Summation. 



IT is often desirable to obtain the approximate value of an integral taken 

 between limits in cases in which, though we can ascertain the value of the 

 quantity to be integrated for any given values of the variables, we are not 

 able to express the integral as a mathematical function of the variables. 



A method of deducing the result of a single integration between limits 

 fr.ua the values of the quantity corresponding to a series of equidistant values 

 ' the independent variable was invented by Cotes in 1707, and given in his 

 Lectures in 1709. Newton's tract Methodius Different ialia (see Horsley's edition 

 .f Newton's Works (1779), Vol. I. p. 521) was published in 1711. 



<'"tes* rules are given in his Opera Miscellanea, edited by Dr Uobert Smith, 

 and placed at the end of his Harmonia Mensurarum. He gives the proper 

 multipliers for the ordinates up to eleven ordinates, but he gives no details 

 of the method by which he ascertained the values of these multipliers. 



Gauss, in his McthoJits nova Integralium Values per Approximationem 

 Inofiiietuli (Gottingische gelehrte Anzeigen, 1814, Sept. 26, or Werke, m. 202) 

 shews how to calculate Cotes' multipliers, and goes on to investigate the case 

 in which the values of the independent variable are not supposed to be equi- 

 distant, but are chosen so as with a given number of values to obtain the 

 highest degree of approximation. 



He finds that by a proper choice of the values of the variable the value 

 of the integral may be calculated to the same degree of approximation as would 

 be obtained by means of doable the number of equidistant values. 



The equation, the roots of which give the proper values of the variable, 

 is identical in form with that which gives the zero values of a zonal spherical 

 harmonic. 



