ON QUADRATURES AND INTERPOLATION. 327 



function of tlie other. That is to say, if one quantity be u, and the 

 other X, it is permissible to write 



^l,^ = rtg + a^x + a^x^ + o„a;" 



when n is the number of intervals, or 91 + 1 that of the values of the 

 function : and then to determine the constants, so that the substitution of 

 the 11 + 1 values x = r^, r^yV^ shall give, for each 



«o = «o + "i^'o + ^"Cj &c. 



There are other equivalent expressions which are sometimes more con- 

 venient, for instance : — 



u.^ = (rtg — cb) (fti — a;) (a„ — a;) 



Assuming a curve to be generated according to a continuous law, 

 if the equation between the ordinate and the abscissa is of a simpler 

 character, when integrally expressed, than 



2/ = ao + aiO! + + «««" 



that will be shown by the as with a high suffix vanishing, or, if finite 

 difierences be used, by the higher differences vanishing. It follows that, 

 if the ordinates are known to be exact, a form may in general be as- 

 sumed, not less simple than the above — that is to say, of a degree only one 

 less than the number of ordinates — and any simplification will appear 

 from the resulting equations. If, however, the law is in reality rnore 

 complex, the assumption made is only good as an approximation. 

 "Whether it be an approximation or not, hinges upon the question of 

 convergence. 



It will be shown further on that, as regards quadrature, the rules 

 run in pairs, 2in — 1 equidistant ordinates giving a result of the same 

 order of accuracy as 2m ordinates, and generally a rather better result 

 arithmetically. This is proved as far as 7 ordinates, and is a probable 

 inference generally. Assuming convergency, the higher rules would 

 seem to be better for the same number of ordinates than the lower. That 

 is to say, the rule of 9 ordinates or 8 intervals is better than the rule of 5 

 intervals or 4 ordinates, taken as 4 + 4, and this again better than the rule 

 of 3 ordinates or 2 intervals taken as 2 + 2 + 2 + 2 ; while this again is 

 better than the polygonal rule taken as 8 x 1 .* The main part of the 

 foregoing reasoning is independent of any supposition as to the equi- 

 distance of the ordinates. But it does imply that the ordinates are 

 exactly given. If this be untrue, it is not easy to see how much proba- 

 bility is involved in our assumption. Apparently this question is indeter- 

 minate, like that of the best mean, where the object of the mean is not 

 stated. Mr. George Darwin has shown f that, for the quadrature of an 

 area of which the ordinates are liable to uncertainty, the broken line 

 simply joining the heads of the ordinates gives a more probable area 

 than that obtained by any of the higher parabolas. 



It may be worth while to remark, that, for the pui-poses of interpola- 

 tion, the relation of ordinate and abscissa is a supposition of mere 

 convenience, not of necessity. The only necessary relation is that of 

 function and variable. 



* It is more convenient to compare the number of intervals into which the base 

 is divided, than the number of ordinates. The arithmetical comparison always turn 

 upon the intervals. 



t See The Messenger of Mathematics for January, 1877. 



