/. H. Alexander on a Ne\o Formula for Interpolations. 23 



It is manifest upon inspection, (and still more by executing nu- 

 merically the ratios,) that a cubic equation developes a series the 

 most nearly approaching that of the distances themselves. 



These two examples, drawn from astronomy and general 

 physics, are sufficient to illustrate the method of interpolations 

 by the present formula, when the terms of the given series are 

 equidistant. I shall now present some cases, which have fur- 

 nished examples already to others; and in which, the ordinates 

 are not equidistant. In point of fact, in a great many physical 

 observations, it is extremely difficult to preserve such isotomy ; 

 and there is hardly ever a case of experiment or observation, 

 though admitting in one of its series a progression by regular mul- 

 tiples, in which all the successive terms can be accepted by the 

 observer, as of equal reliability. The application of the formula 

 to broken series of this kind, is, therefore, of the highest interest. 



3. An instance may be taken, already cited by Wallace, where 

 the distances of a comet from the sun's centre, were as follows, 

 at the given epochs, viz. : 



At midnight on December 12, the distance was 301 ; 



" " 21, " " 620; 



« 24, " " 715; 



« 26, " " 772; 



and he has applied the ordinary method for interpolating the dis- 

 tance on 20th December at midnight. I shall use for the same 

 purpose, the formula III. 



In order that the first term of the series for epochs should be 

 unity, the origin of the co-ordinates has to be taken as of 11th 

 December ; and the two series will then correspond as under : 



Epochs of time-intervals : 1. 10. 13. 15. 



Distances : 301. 620. 715. 772. 



The epoch to be interpolated corresponds, then, to 9 in the series ; 

 but for the application of the formula, we have first to ascertain 



the values of 6, c, and d corresponding to the epochs 2 ; 3 ; 4, 



i 



respectively; the value of a is already known to be 301. For 

 mis purpose we construct with the three other distances given, 

 as many equations as there are unknown quantities, substituting 

 ln each the successive values of n as 620 ; 715 ; 772, respectively, 

 as under; and then eliminating the quantities in question : 



«= 1 0. j and 620 = 'hSJ. d - h**. c + ^1^. 6 - ^L- 6 . 301; 



2.3. 2 2 2.3. 



«=13. ; and n6 J*£™d-B™. c+ U }™.l,- * "^.SOl j 



2.3 2 \a 2.3 



n=15,a„d772= 1 A^. rf Ji4H l , + 1 i:^ 1 .6-l^i 2 i 1 .301 ; 



2.3 2 2 2.3 



