OK APPROXIMATE MULTIPLE INTEGRATION 



Uteo nce the form of the function 11, and therefore the values of the coefficients 

 Ac. must be considered entirely arbitrary, we may equate the 

 coefficient* of A., Ac. in equations (11) and (13) as follows: 



. j/tt'+Ac.-ifl'+iy =B lt 



' 



X 

 2^c,*+ 



Ac. 



If we write S t for the sum of all the values of *, 



S, for the sum of all products such as ,*, &*, 

 S, ................................................ f, f, 



then for r terms 



B t S r - BjS,_ , + 5A. , - &c. ( - Y B r+l = 0, 

 B& - BJS^ + B t S,_, - &c. ( - )' B r , t = 0, 



a aet of r equations, from which the quantities JR., .R, have been eliminated, 

 ami from which we may determine the r quantities S, . . . S r , and the values 

 <f are then given as the roots of the equation 



r-s,r-'+s 8 r- 4 -&c. (-) r s r =o. 



Thus if we have three values of they should be 



' 



4=0, 



When the quantity to be integrated is a perfectly general function of the 

 variables we must proceed in a different manner. 



