

J. H. Alexander on a New Formula for Interpolations. 15 



dity of my method, I suppose it will not be an unwelcome addi- 

 tion to our existing analytical apparatus. It may be so expected, 

 especially at this period, when there is less lack of observations 

 in the physical sciences than of elegant, or even expert, methods 

 in their subsequent employment ; and, in so far, what follows may 

 be considered as a contribution to a fruitful and extensive part of 

 the Art of Observing. #% 



Interpolation is, in general, analytically the supplying of miss- 

 ing or required quantities within or beyond (either positively or 

 negatively) a series whose given quantities progress according to 

 some certain, but unknown, ratio. When the ratio is known, it 

 is of course more convenient to find the required quantities by 

 the equation it furnishes. Such quantities, if numerical, are (or 

 are assumed for the purpose of interpolation to be) terms in a 

 series of figurate numbers ; and the order of the series, as well 

 as the equation by which it is generated, is determinable by the 

 number of removes after which the differences of the successive 

 terms become constant, or, in most practical cases, by the actual 



number of terms given in the series. In geometry, a similar ap- 



plication of it is made for the determination, by rectangular co- 

 ordinates, of a line subject to the condition of passing through a 

 given number of points. t 



In this last aspect, if the number of points given be no more 

 than two, it is manifest that the conditions of the question are 

 met by a straight line. If the points are more than two, and the 

 construction does not shew the co-ordinates for such points to be 

 in the relation of the base and perpendicular of a right-angled 

 triangle, the development of the line required must be by a curve 

 traced from point to point by means of an ultimate reduction to 

 this same relation. It is plain that such curves may be as varied 

 as the number and positions of arbitrary points given for their 

 determination ; but the order to which they belong and the char- 

 acter of the curve itself, follow from the degree and co-efficients 

 of the equation which is found to express the respective values 

 of its abscissa and ordinate. Thus in all the curves determined 

 by normal sections of an equilateral cone, the equation for the 

 ordinates is of the second degree ; all arcs of a circle, therefore, 

 are lines of the second order, and so are those of an ellipse and 

 common parabola ; but the character of the former of these is 

 still farther defined by the co-efficient expressing the ratio be- 

 tween its transverse and conjugate diameters, and of the latter, by 

 the co-efficient of its parameter. The cubic parabola, however, 

 still coupled with a similar co-efficient, presents an equation for 

 ordinates of the third degree, and is therefore a line of the third 



order. ^ *** * 



I may remark here, in passing, that writers on analytical ge- 

 ometry have not unanimously retained these distinctions in view ; 





