ON QTJADRATUBES AND INTEKPOLATION. 363 



polation to the direction of a normal line or tangent plane. This, in the 

 ordinary language of partial differential coefficients, is the evaluation of 



^{l+f + q']. 



The first thing is to find p and q by the methods for interpolation of 

 ordinates (IV, 6) and then to form the radical. Graphical methods are, 

 however, generally the most convenient for this. The integration of this 

 radical in two dimensions gives the surface. For an example of this 

 particular form of interpolation combined with quadrature, see Scott 

 Russell's ' Modern Naval Architecture,' and the 'Transactions of the 

 Institution of Naval Architects,' vol. vi. for 18G5, p. 64. 



Tables of treble (or higher multiples) entry are of course confined 

 within very narrow limits. Their interpolation calls for no special re- 

 mark, unless it be that the indeterminateness, as well as the complexity, 

 increases with every additional dimension. This remark is true, as a 

 separate consideration, firstly with regard to the general indeterminate- 

 ness of all interpolation, and secondly with regard to the indeterminateness 

 of the inverse processes, with reference to the possibility of more roots 

 than one, and to the requirement of more data than the ordinate.* 



V. — Inteepolation and Quadeature with Ordinates not equidistant. 



Section 1. — Newton's method. 



The principal theorem of interpolation when the distances between flie 

 ordinates are arbitrary instead of equidistant, is given by Newton, in his 

 ' Principia,'t under this title, ' Invenire lineam curvam generis parabohcl 

 quae per data quotcumque puncta transibit.' It consists of a method of 

 divided differences, and Newton uses it in a form which is, as exactly as 

 may be, the counterpart of that which he uses when the ordinates are 

 equidistant. It is not thought necessary to reproduce it here, as the 

 reference to it is very easy. J 



Professor Emory McClintock, of Milwaukee, has given § a modifica- 

 tion of Newton's formula, which lends itself better to logarithmic com- 

 putation. The terras being <l>Xi or ^o^^i) ^^-x or ^o^2> ^Jid so forth, the 

 general term of his divided differences is 



,L ™ <^m ^n — */*». ^.,.+ 1 



which gives, on multiplying up, 



<Pm ^n — <t>m <«m+l + («n " Km+l) fm+l- ««• 



Giving m the values 0, 1, 2, &c. in succession, and substituting suc- 

 cessively, we get 



^0 2J„ = <poXi + (x„ — a;,) ^1 a;2 + (x„ — x^) (x„ - x.^) <t>o x^ 



+ («« - ^l) « - «2) (»« -X3) f3Xi+ 



* See on this G. Darwin, on ' Fallible measures of variable quantities,' London, 

 Edin. and Bvblin Philos. Mag. for July, 1877. See also Lacroix, Traiti de Calcul, 

 vol. iii. p. 44 et seq. (arts. 913 et seq). 



t Book iii. Prop. xl. Lemma V. case 2 ; see also his Methodvs Differ entiaUs, Prop. iv. 



X See also De Morgan, Calculus, p. 550; Lacroix, Traitc de Calcvl, vol. iii. p. 31, 

 et seq. — or page 552 of the Cambridge translation. See also Stirling, Interjwlatio 

 Scrierum, Prop, xxix., and Hall, 'Finite Differences,' Encycl. Metrop. p. 249. 



§ See the American Journal of Mathematics, ^ure and aiyplied, vol. ii. pp. 307-314, 



