1877.] Integration between the limits of Summation, 41 



then since the form of the function u, and therefore the values of 

 the coefficients A^, A^, A^, &c. must be considered entirely arbi- 

 trary, we may equate the coefficients of A^, &c. in equations (11) 

 and (13) as follows : 



B, + 2E^ + 2R^ + &c. =1 =i?^, 



m,K,' + 2i2,C + &c. = I /S^ + ^ /3V + J y =.B,, 



If we write 8^ for the sum of all the values of ^^, 



8^ for the sum of all products such as ^^, \^^, 

 8 . V V t:^ 



then for r terms 



S^S^ - J^A-^ + BA-. - &c. (-)"5^,, = 0, 

 • BA - BA-r + B,8^_, - &c. (-)'-5,,, = 0, 



Br,A - Br,A-r + Br,A-. - &c. i-YB,.^, = 0, 



a set of r equations, from which the quantities B^, B^ have been 

 eliminated, and from which we may determine the r quantities 

 8^... 8^, and the values of ^ are then given as the roots of the 

 equation 



Thus if we have three values of ^ they should be 



?o = 0, ?i = ± [3^^ + lO^y + 3y]4 [5^^ + 57^]-^ 



_4 /3^+5/3y + 7 ' p _5 ^+2/3y + 7^ 



° 3 3/8*+10/8y + 37*' ^ 6 3y8^+10;6y + 37*" 

 Vol. III. Pt. il 4 



