Of HJaUJLimTir MULTIPLE rNTEGRATION. 



Thwi if we endeavour to determine twenty-seven sets of values of x, y 

 mad i, with a uunMpnnrtinff * of multipliers, so as to express the value of 

 U* triple integral in the form 



P f f 



......... (l), 



when* . denotes the value of u when x = y = z=Q; 



S(M ) iluootoi the sum of the six values of u for which 



x= pa, 0, 0, 



y=o, p d, o, 



z=0, 0, pc-; 



2( f ) denotes the sum of the twelve values of u for which 



x = 0, qa, qa, 

 y=qb, 0, qb, 



* = qc, qc, 0; 



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



x= ra, 



y= rb, 



z= rc, 

 where the signs may be taken in any order. 



The equations to be satisfied in order that equation (1) may be satisfied 

 for any function of x, y and z of not more than seven dimensions are 



+ 6P +12Q+8R =8, 



