8 PROCEEDINGS OF THE AMERICAN ACADEMY 



-A+B A+B B-G B'+C 



m. s. m. e. s. e. 



—6 18.3465 6 19.8119 0.1522 0.2156 



—6 21.8330 6 24.4922 0.2910 0.3748 



With these values (6) gives 



R. A. A 1 A 2 



: 



The following very convenient method of interpolation was published 

 several years ago in an imperfect form in the Mathematical Monthly. 

 The expressions for the constants have been here obtained in a quite 

 different manner, and are given in a different function of the differ- 

 ences, which are more exact, and the whole matter is given in an 

 improved form. 



If we put in (1) 



(8.) F X = F + B 1 x+B i a* + B 3 x% 



and determine -^, B 2 , and B s , so as to make the expression coincide 

 with (1) when x = %, x = §, and x = 1, we obtain very nearly, re- 

 instating A 1 — £ A 2 for A/, A 3 — £ A 4 for A/ 5 , &c, 



B^" • • •$ 



US 



The greatest possible error, E, in using (8) with these constants 

 instead of (1), is only 



(9.) E = ± UVtt A 4 + TT W A 5 + T Ao A 6 ) 



