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



just as, geometrically, two fixed points are enough for determin- 

 ing the position and direction of a straight line. In a series of 

 the third order, it is necessary to use four terms; and in one of 

 the fourth order, five terms, for its development. 



The choice, however, in any case, of the proper number of 

 terms to be involved in the calculation, is not arbitrary ; but de- 

 pends in each instance upon the character of the series in ques- 

 tion. Where such a series corresponds with equidistant ordi nates, 

 in general successive differences of its terms may be taken ; and 

 the degree of removal where such differences are constant, is the 

 degree of the equation applicable and the order to which the 



series itself belongs. Thus if we have corresponding to 



the ordinates : 1". 2". 3". 4". 5". &c., 

 a series: 16 ft. 64 It. 144 ft. 256 ft. 400 ft. &c., 



(whose terms, in fact, represent the spaces passed over by falling 

 bodies in the respective times) we shall have also 



1st differences 48. '80. 112. 144. <fcc. 

 2d " 32. 32. 32. constant. 



The equation of this series is of the second degree, then ; and 

 any three terms of it, are sufficient for the interpolation of the rest. 

 So, if we take the series of expansions of iron, (whose length 

 is 1.— at 0° Centigrade,) corresponding to the temperatures indi- 

 cated respectively by the 



ordinates : 100 J C. 600° C. 1 100° C. 1600° C. 2100° C. &c 



expansion) 00114 001148 0-03006 005751 009446 &c. 

 series: ) 



we shall have, 



1st differences : 1034. 1858. 2745. 3695. &c. 



2d « 824. SS7. 950. &c. 



3d « 63. 63. constant. 



Its equation, then, is cubic ; and four terms are necessary to inter- 

 polation of other terms of such a series. 



The same conclusion may be arrived at by other considera- 

 tions, sufficiently probable to serve as guides. As the ordinates 

 actually expressed in degrees Centigrade are equidistant, they 

 may be transformed into the natural series, having its first term 

 at 100° C, thus 1, 2, 3, 4, &c. ; and we can then judge what 

 power of this natural series will correspond the nearest with the 

 actual expressions. The presentment may be conveniently made 

 in the following form : 



Actual ordinates : 100° 600° 1100° 1600° 2100° 



Transformed terms : 1 2 3 4 5 



Squares: 1 4 9 16 25 



Cubes: 1 8 27 64 125 



Fourth power: 1 16 81 256 626 



Actual experiments di- f 1 m rv or or za ac * 



vided by 1st term : \ l 1(H) ' 2(r37 5 ^48 &c. 









i 



