ON QUADEATURES AXD INTERPOLATION. 365 



. Gauss's proposal is to select the absciss© in sucli a manner that the 

 error of a quadratui-e, obtained by means of n selected ordinates, shall 

 disappear in the application of this quadrature to any rational function 

 not exceeding the degree of 2 n—l. Tor this purpose, supposing the limits of 



integration to be ± 4' the abscisste are the n roots of the equation 



dx" \ 4: J 



which are the halves of the roots obtained by equating to zero the co- 

 efficients of Legendre.* Thus the multipliers for quadrature are easily 

 obtained by substitution or by indeterminate coefficients. The roots of 

 the sets up to n = 5 can be obtained as quadratic surds, and, with the co- 

 efficients, are as follows : 



11 =■ 1, X = 0, Ci = 1 



O o 1 1 



11 =2, x-= j^, Ci = C2 = — 



_ 5 _ 4 



11 =i, 0=2 = :^ (15 ± 2 V 30) 



140 



4 ^ 72 



C2 = C3 = — + ;^^ -v^ 30 



n = 5, a;2 = ^ (35 ± 2 ^ 70) or a; = 



Ci = C5 = 



252 



d22-lB^/_70 



1800 

 322 + 13 n/ 70 



1800 

 _ 6J^ _|. 64 . 

 "^ 1800 225' 



Section 4. — OtJiei- Methods and Supj)odtwns, 

 Tor some extensions of Gauss's method, and for some particular forms 



* See Legendre, Fonctions UUyMqves, vol. ii. p. 531 ; Todhunter on the Functions of 

 Laplace, S'c., chapter s. ; Boole, Finite Differences, 2nd edit. p. 51, (ch. iii. art 12) ; 

 Bertrand, Calcul Integral, p. 339. Gauss's own memoir is Met/wdvs nova integralium 

 valores 2>er ajjproximationeminveniendi. Gottingen — Comm. III. [ISli] JVouv. Ann. 

 Math. XV. (1856). 



t The numerical values of these and of some of their logarithms are given by 

 Gauss. Bertrand, and Todhunter in the works already quoted. The limits of error 

 are also fully discussed by the two latter. But an attentive perusal of Bertrand's 

 reasoning will show that the Umit of error depends upon the convergency of the 

 parabolic expression of the function, and cannot be relied upon where this is not 

 secured. 



