METHODS OF INTERPOLATION. 34^ 



then the four-term formula (b) gives — 



and substituting these values in formula (78), and taking — 



N 4 

 we get the equation — 



u = 8.25 + i sin 45° { — 5 sin {x 6) + cos {x 0)\ — 2.25 sin 2 {x 9) 

 which is transformable into — 



u = 8.25 + 1.803 sin {x + 168° 41') - 2.25 sin 2 {x d) 

 When the values — |, — ^, + ^, + f , &c., are successively assigned to 

 X in this equation, the resulting values of it will be the four given terms 

 or data, and for any assumed intermediate values of x intermediate 

 values of the function can be interpolated. 



One of the formulas of the Repertorium above mentioned, the one 

 which includes twelve given terms, is the same as the formula subse- 

 quently demonstrated and used by Everett^ with a different notation, in 

 his articles on "Eeducing observations of temperature," in the Amer- 

 ican Journal of Science and Arts for January and September, 1863. In 

 connection with the remarks made at page 314 of my previous memoir, 

 respecting Everett's method of correcting annual equations of tempera- 

 ture, it ought perhaps to be said that I have not had access to the Edin- 

 burgh ISTew Philosophical Journal for July, 1861, in which his work was 

 first published; but the method is presumed to have been essentially 

 reproduced by him in the American Journal for January, 1863. (See 

 foot-note on page 27 of the latter.) 



INTEEPOLATION BY MEANS OF AN EXPONElfTlAL FUNCTION. 



In the discussion of this subject at page 329 of the previous memoir, 

 all the roots of the equation of relation were supposed to be real and 

 positive. We shall now consider the general case in which the roots 

 may be either real or imaginary, and shall proceed as before, after the 

 analogy of Prony's treatment of the subject of ordinary interpolation 

 from single terms or ordinates. Instead of placing the origin of co- 

 ordinates at the middle of the left-hand group as before, we shall now 

 place it at the middle of the series. This is equally convenient, and 

 more symmetrical and accordant with the system we have followed in 

 the cases of algebraic and circular functions. 



The appearance of imaginary roots in the equation of relation implies 

 a change in the form of the function whose equidistant values constitute 

 the recurring series. Just as each real and positive root corresponds to 

 a term of the form h iS"" in the function, so each pair of imaginary roots 

 corresponds to a term of the form — 



\G sin (x 6) + d cos {x 0) \ y'^ 

 where c, ^, and y are constant numbers, and ^ is a constant arc. ISToWy 

 let us write the equation of the curve as follows: 



