OX ArWUMCIMATS MULTIPLE INTEGRATION 



There are -l equations in the second group, in which = 2, 



4 



(21)73 ' 



There will be a- 3 equations in which = 4, and so on. Hence if a is 

 the whole number of equations is 



(H- 



If a is odd, the number is 



4 



To satisfy these equations we have in general, for each group of values 

 of ii, three disposable quantities, R, p and q. 



If, however, the central ordinate be selected it will constitute the first 

 group, and will introduce only one disposable quantity, namely JR t . 



Also, if ordinates lying on the axes of p or of q be chosen, the groups 

 so formed contain only two disposable quantities, one of the ordinates being 

 zero. 



Also, for ordinates lying on the diagonals, q=p, so that for these also 

 there are only two disposable quantities. 



Thus if a = 3, the number of equations is - - = 6 ; and if we select the 



central ordinate, giving one disposable quantity, a group of four points on the 

 axes, giving two disposable quantities, and a group of eight points giving 

 three disposable quantities, we shall be able to satisfy the six equations, and 

 to form an expression for the integral which will be correct for any function 

 not exceeding the seventh degree. 



We assume 



n n 



b r) W 







