50 



REPOET 1879. 



P 2 C*0 = K3.-« 2 -i), 



F\x)=±(5x 3 -3x), 

 P*(as) = |(35a5 4 -30a; 2 + 3), 

 V 5 (x) = ^(6Sx 5 -70x 3 + 15x), 



v 6 (x) =TV(' 231a!6 - 315a!4 +i 05a!2 - 5 ). 



P7(aj) = T V(429z 7 -693;e 5 + 315:e 3 -35a;). 



The functions present themselves extensively in the higher parts of 

 mathematics (in reference to the attraction of spheroids and other physical 

 theories l ) ; but they first occur in the theory of interpolation : see Gauss, 

 1 Methodus nova integralium valores per approximationem inveniendi ' 

 ('Comment. Gott. recent,' t. iii. pp. 39-76 (1816), or ' Werke,' t. iii. pp. 

 165-196), from which the numerical results given in the present intro- 

 duction are taken. 2 



Suppose that y, a function of x, has to be approximately determined 

 for the range x = to x = 1, by means of the values of y corresponding 



to n given values of x over this range ; or say that the integral / ydx 



has to be thus determined. In the original theory, as developed by Cotes 

 in the 'Harmonia Mensurarum ' (1722), the given values of x are taken 

 to be at equal intervals, viz., for n = 2, they are 0, 1 ; for n = 3, they are 



Representing y as a function of x of the order n — 1, and determining 



1 See Todhunter's Treatise on Laplace's Functions (1875), Ferrers's Treatise on 

 Splwrieal Harmonics (1877), or Heine's Handbuch der KugelfwncUorwn (1878). 



2 A short notice of Gauss's method is given in Boole's Finite Differences, second 

 edition, edited by Moulton (1872), ch. iii., art, 12, pp. 50-53. 



