BETWEEN LIMITS BY SUMMATION. (J07 



We may begin as before by transforming the double integral into one 

 between the limits +1 for both variables, so that 



=1 rudxdy = (x. i -x 1 )(y,-y 1 )\ i udpdq .............. (l). 



J *J Vi J -IJ -1 



Let () denote the sum of the eight values of u corresponding to the 

 following eight systems of values of p and q, 



(a n , &), (a n , - b n ), ( - a n , b n ), (-a n , - b n ) ; 

 (b n ,a n ), (b n ,-a n ), (-b n , a n ), (-b n> -a n ), 

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



/=(*,- x,) (y, - yO^K) + R^( Ul ) + &c. + J2.2K)} ......... (2). 



The values of the coefficients JR, a and b are to be deduced from equations 

 formed by equating the sum of the terms in p-<f in this expression with the 

 integral 



Only those terms in which both a and /3 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 a or /3 is odd must disappear. 



Also since the expression is symmetrical with respect to p and q, the 

 term in yf(f will give an equation identical with that in p*q ft . 



We may therefore write down the equations at once, leaving out the factor 



p"q f 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 

 0-0, 



=4, 



] =f, 



=f, 



