ON MATHEMATICAL TABLES. 51 



the coefficients in this manner, we have an expression for y from which 

 the integral / ydx may be calculated. Denoting the interval by a, that 



is, writing a = ^Z\> tbe resulting formulas, corresponding to the values 

 n = 2, 3,... 11 respectively, are as follows : — 

 Thus, for example, 



or =iY + |Y+iT 1 , 



or =iY + fT 4 +tT J + iT 1> 

 &c, &c. 



In the new theory of Gauss, it is shown that it is advantageous to take 

 the given values of x not at equal intervals, but to be the values which 

 are the roots of the equation 



P»(2aj-1) = 0; 



thus f or n = 1 the value is x = i for n = 2 the values are i ± o-7o> 

 and so on. 



The resulting formulas are as follows : — 



/ ydx = Ay a if n = 1, 



= Ay a + A'y af ]£n = 2, 



= Ay a +A>y a ,+A"y a ,,i? n = 3, 



= &c, 



where the values of a, a'... and the coefficients A, A'... for the different 



values of n are 



n = 1, 



A = l. 



Approximate correction = ^ L". 1 

 to = 2. 



1 Suppose in general the true value of y is 



V = L + L'(^-i) + L"(«-i) 2 + &c, 



then the correction to be applied to / ydx in the general case is 



where Z« denotes the correction to be applied to / {x-±)™dx; so that Z (2nl L (2n1 



t/ o 



may be regarded as the approximate correction to / ydx. Thus, for example, y 

 being as above, 



f\dx= L + i L" + il/v + 4 _i_l- + to. j 

 if » = 1, the formula gives L, and the approximate correction = v 9 L"; if n = 2, 

 the formula gives L + ^ L" + j|j L iv + &c, and the approximate correction 



E 2 



