1877.] Theory of Attraction. 89 



density would be infinite. Hence if the whole quantity of fluid 

 is finite it must be accumulated entirely on the surface, and the 

 interior will be entirely empty, as we know already. 



If the force is inversely as the fourth j)ower of the distance the 

 density within the ellipsoid will be uniform. 



(2) Prof. J. Cleek Maxwell. On Approximate Multiple 

 Integration between Limits hy 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 be- 

 tween limits from the values of the quantity corresponding to a 

 series of equidistant values of the independent variable M^as 

 invented by Cotes in 1707, and given in his Lectures in 1709. 

 Newton's tract Methodus Differentialis (see Horsley's edition of 

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



Cotes' rules are given in his Opera Miscellanea, edited by 

 Dr Kobert Smith, and placed at the end of his Harmonia Mensu- 

 rarum. 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 Methodus nova Integralium Valores per Approxi- 

 'mationem Inveniendi (Gottingische gelehrte Anzeigen, 1814, 

 Sept. 26, or Werke, ill. 202) shows how to calculate Cotes' multi- 

 pliers, and goes on to investigate the case in which the values of 

 the independent variable are not supposed to be equidistant, 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 double 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. 



Double Integration. 



There is a particular kind of double integration which can be 

 treated in a somewhat similar manner, namely, when the quantity 



