42 Prof, Maxwell, On Approximate Multiple [Mar. 12, 



If/3 = 7 = 1> 



^0-12' -"i 24' 



When the quantity to be integrated is a perfectly general 

 function of the variables we must proceed in a different manner. 



We may begin as before by transforming the double integral 

 into one between the limits + 1 for both variables, so that 



/ =j'''j\dxdy =^{x,- a^i) {y, - 2/i) J J ^''dp^^. (!)• 



Let S {uj denote the sum of the eight values of u correspond- 

 ing to the following eight systems of values of p and q, 



K K)> K - ^n), (- a„, Kl (- ««' - K) ; 



and let us assume that the value of the integral is of the form 



r=l(^.- ^i) (2/. - 2/i) f^o^; w + ASK) + &c. + B^%{u:)] ... (2). 



The values of the coefficients R, a and h are to be deduced 

 from equations formed by equating the sum of the terms in j)"-^^ in 

 this expression with the integral 



0V/4«^2 = (, + i)*^ + l) (3). 



Only those terms in which both a and yS are even will require 

 to be considered, for the symmetrical distribution of the values of 

 p and q ensures that the terms in which either ct or /3 is odd must 

 disappear. 



Also since the expression is symmetrical with respect to p and 

 q, the term in p^q"^ will give an equation identical with that in 

 p'^q^. 



We may therefore write down the equations at once, leaving 

 out the factor p'^q^ common to each term, but writing it at the side 

 to indicate how the equation was obtained. There are a + 1 

 equations in the first group, in which /3 = 0, 



