1877.] Integration between the limits of Summation. 



45 



Triple Integration. 



lu extending this method to triple integration we meet with 

 the curious result, that in certain cases the solution indicates that 

 we are to employ values of the function some of which correspond 

 to values of the variables outside the limits of integration. 



Thus if we endeavour to determine twenty-seven sets of values 

 of X, y and z, with a corresponding set of multipliers, so as to 

 express the value of the triple integral in the form 



r [ [" udxdydz = abc[Ou^-\-Pt{u) + qZ{u^) + E%{u)].,.{l), 



where u^ denotes the value of u when x = y=z = i); 



S (Wp) denotes the sum of the six values of u for which 

 x= ± pa, 0, 0, 

 3^ = 0, ±ph, 0, 

 ^ = 0, 0, ±pc; 

 % (u^) denotes the sum of the twelve values of u for which 

 x = 0, ±qa,± qa, 

 y=±qh, 0, ±qb, 

 z — ±qc,±qc,0\ 

 and 2 (mJ denotes the sum of the eight values of u for which , 



x= ±ra, 

 y = ±rb, 

 &= ±rc, 

 where the signs may be taken in any order. 



