ox QUADRATURES AND INTERPOLATION. 335 



I («_2 + «2) = «o + 2'«2 + 2^a4 + 2««« + 

 H («-3 + ^'3) = tto + 3^a2 + 3%4 + 3«a6 + 



Now, introducing indeterminate mnltiijliers, 



i-Xo + X. + Xo + X3 + ^n^n 



X, + 22X0 + 32X3 + . . . . n%, = In^ 



X, + 2^X^ + 3^X3 + . . . . n%, = ^ n^ 



X, + 26A2 + 36X3 + . . . . n%, = Li^ 



• •••••«••• ^— • « • • 



from whicli the value of any X is easily formed, by means of determinants 

 if necessary. For any given value of 01 the coefficients are those of the 

 corresponding rule for 2 n ordinates or 2n + 1 intervals. Thus, stopping 

 at n ^ 1 



^ Xq + Xi = 1 , Xi = ' .*. Xq = ^ 



/ 



1 1 



Udx := - (m_i + 4i<o + Ml) 

 _1 o 



Again, stopping at m =2 



|Xo + ^1 + ^2 =2, Xi + 4X2 = I Xi + 16X2= ^ 



, . 14 , 64 . 24 



whence X2 = ^,Xi=-,Xo=^ 



udx =-f^ (7«_2 -V 32m_i + 12?(o - 32mi + 7«2)- 



2 45 



/ 



The rules for an odd number of intervals or an even number of 

 ordinates may be got by giving x the successive values ± 1 , ih 3 , + 5 



Then «_„ + ?<„ takes the same value as before. Writing 



n = 2m, + 1, and using as indeterminate multipliers yuj, /U3, /itj . , . . the 

 equations become 



^'1 + /^3 + ^'5 + /'2»,+i = 2m + 1 



^i + 3V3 + 5V5+ (2 m + 1) V2.+1 = ^ (2 «i + 1)^ 



^1 +3^3 + 5^5+ (2 m + l)V2m+l = ^ (2 w + 1)5 



Stopping at m = 0, /j 



/- 



1 I 



udx = - (m_i + «!) 



