20 J. H. Alexct7ider on a New Formula for Interpolations. 

 IV. "When fourth differences are constant, 



n-l.n-2.rj-3.7i-4 n-l.n-2.n-3.w-5 n -l.n-2.n-4.n~5 

 1.2.3.4 '* 1.2.3.1 "* 1.2.2.1 



n-1 .n-3.n-4 .n-5 n-2.n-3.n-4.n-5 



FT 2 .3.1 ' ^ 1.2.3. 4 ' a 



Such are the formulas covering a case where as high asfve 



terms of the original series, concur for the interpolation. It 

 would be easy to extend them further ; but the occasion for it 

 will hardly occur in practice. I have left them in their present 

 shape, in order to show more clearly the mode of their generation 

 and their symmetry ; besides, this shape will also be found as 

 convenient for calculation. A transformation, however, can very 

 easily be made ; that may be preferable for those who make the 

 entire calculation, for greater security, upon paper. The second 

 series, for instance, will then assume the form ; 



n*-3w+2 n 9 -5n+6 

 II. l~ q . c — (n 2 — 4ra + 3). 6+ o -a. 



2 



2 



farther mere 



elementary processes. 



It may be remarked that the form of these several series is ap- 

 parently divergent; they are not less applicable, however, to 

 series that converge. The change of signs follows such char- 

 acteristics, and unerringly regulates the result. 



I shall now offer in proof, some illustrations of the application 

 of the formulae to practical cases ; in which there will be an op- 

 portunity of comparison with the methods of interpolation ordi- 

 narily employed. 



1. Weisbach, in his learned essay on Observations,* has given 

 the following series of temperatures and pressures, furnished by 

 certain experiments on steam ; viz. 



Pressures in atmospheres : 1 1-5 2 2-5 



Temperatures (Centigrade): 100° 112° -2 122° 129°; 



•Allgemeine Maschinen Encyclopadie; von Hulssc u. A. Art. Beobachtung. 







and desires then to interpolate the temperatures proper to pres- 

 sures of 1-1; 1-2; 1-3; 14 atmospheres, respectively. He as- 

 sumes, also, that the differences in the temperature-series, are 

 constant in the second place ; and, therefore, employs in fact but 

 the first three terms. As my object is only to compare my pro- 

 posed method with th« habitual one under the same assumptions, 

 I shall not stop now to discuss the propriety of these ; but shall 

 give at once the numerical substitutions in my formula. 



In arranging the equidistant pressures: 1. 1*5. 2. 2 5; &c. 

 according to the natural series for or- > n ~ A £ 

 dinatcS, viz. : * L % 3 ' 4 : &c " 







i 



