1877.] Integration between the limits of Summation. 43 





There are a — 1 equations in the second group, in which 



/S = 2, 



_p2a-222 ^ [^.P.V»^(P^-^ + g^-^)] = ^2a -1) . 3 • 



There will be a — 3 equations in which /3 = 4, and so on. 

 Hence if a is even, the whole number of equations is 



(i-^y- 



If a is odd, the number is 



(ct + l)(a + 3) 

 4 ' 



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

 of values of u, three disposable quantities, R, ^ and q. 



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

 the first group, and will introduce only one disposable quantity, 

 namely R^. 



Also, if ordinates lying on the axes of _p or of g 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 = Pi so that for 

 these also there are only two disposable quantities. 



.4.6 

 Thus if a = 3, the number of equations is —^ — 6 ; and if we 



select the central ordinate, giving one disposable quantity, a 



4—2 



