TRANSACTIONS OF THE SECTIONS. 



13 



chaque point dii jilan coiTesponde a uiie droite unique de la coupruouoe. iSi Ton 

 tninslbrme le plau par les polaires reciproques de Poucelet, les images des droites 

 de la cougi'ueiicc seront les droites du plan repr^sentatif. 



0)1 Graplilcal Interpolation and Integration. 

 By George H. Daeavix, M.A., Fellow of Trin. Coll., Cambridge. 



Suppose a number of points A, B, C, &e. are given on equidistant ordiuates of a 

 cur^'e, and that it is desired to draw a curve tlircugh them. This ma}^ be best 

 done by interpolating intermediate points. I^et a, b, c, &c. bj ordinates halfway 

 between A and B, B and C, &c. Join AB, BC, &c. along the whole curve, and let 

 the ordinates a, b, c, &c. intersect AB, BC, CD, &c. in/, (/, h, &e. Join AC, BD, 

 CE, &c., and so on along the whole curve, and lot the ordinates B, C, D, &c. inter- 

 sect AC, BD, CE, &c. in F, G, II, &c. Then the rule for interpolating on the 

 ordinates a, b, c, &c. is: — produce afto I, and make// = \ BF ; produce bg to m, 

 and make gin = 5 CG, and so on. In carrying this out practically, several of the 

 above lines need not actually be drawn. 



This rule may be proved from the properties of the circle of curvature, which 

 passes through three consecutive points, such as A, B, C. It gives results correct 

 as far as second differences. A slightly different result will be obtained by working 

 along the curve in the opposite direction : to obtain a better result work both ways 

 along the curve, and choose the points which lie halfway between the discrepant 

 readings. The residt so given is correct as far as third diffei'ences. 



In determining the approximate value of a definite integral it is often convenient 

 to find a geometrical construction for giving a line proportional to the function to 

 be integrated, and then to determine half a dozen values of the function. But the 

 question then arises as to how these terms are to be combined, so as to give the 

 required integral — whether by the rules given by the calculus of finite differences, 

 or by the simpler rule of taking the mean of the extremes and adding it together 

 with all the rest, and multiplying by the common difference. Each ordinate or 

 term is affected by an error, and it may be that the theoreticaUy best rule may give 

 a higher probable error to the result than the more imperfect rule. If, for example, 

 we have seven ordinates, each subject to a probable error c, Weddle's rule (see 

 Boole's Calc. Fin. Diff.) would give a result subject to a probable error 2-846 he, 

 whilst the worse rule only gives a probable error 2-.34o he, where h is the common 

 difference. It must therefore remain indeterminate whether more is gained by a 

 diminished probable error or by a better rule of quadratures. The question could 

 only be determined by some knowledge of the amount of probable en'or of each 

 ordinate, and of the abruptness of the curvature of tlio curve*. 



On certain Determinants. Bg J. W. L. Glaishef, 3J.A., F.R.8. 



The author gave the folloAviug results : — 



I. If P„ denote the number of partitions of n into the elements 1, 3, 8, 4, 

 repetitions not excluded, then 



P = 



(« rows) 



where the first column is 



* The paper is printed in cxtcnso in tlio ' Messenger of Mathematics,' Tol, ri. (January 



Ol(). 



