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



* 



assign to a, b, c, and d, the values of the shadow-lengths, count- 

 ing from 22 June ; and execute the numerical transformations as 

 before, we shall have the terms to be made equal to zero, as 



under : 



Jn 3 +4n a -21* n + 31776 = 0. 



Differentiating, we have 



w 3 + 16 n = A2i 

 and . n =2-31: which counting from 23 June, 



corresponds with June 20 d , 16 h , 33 m -6, as before. 



The epochs obtained by Wallace, are, calculating by 



1. the first, second and fourth terms, June 20 d , 17 b , 15 in : and 



2. the first, third and fourth " " 20^, 17 h , 25™. 

 But I cannot consider the concurrence of these partial methods 



as comparable with a result from a general solution. 



In point, of fact, the present formula admits, from its geometric 

 construction, of being applied to cases like this, with results ap- 



proximate enough for most practical purposes and without any 

 aid from the higher analysis at all, — a contingency where the 

 ordinary formulas fail. For instance, it is plain from the march 

 of the terms just now given that they indicate points in the two 

 arms of a parabola whose apex is the very solstitial point in ques- 

 tion. In reality, these are not in the two arms of one parabola; 

 but in the opposite arms of two parabolas of different parameters, 

 whose axes and poles coincide. This condition, however, only 

 affects the process, not the principle. 



As we have only two points in either arm, we cannot interpo- 

 late nearer than by first differences that determine only a straight 

 line — a path very variant from the hypothetical motion of the 

 sun or the real motion of the earth. But as this motion is ellip- 

 tical, and therefore subject to equation not higher than of the 

 second degree, we may hence derive an approximate use of these 

 differences to serve in solution. 



Starting, then, on 19th June, with the series of shadow-lengths 

 as abscissae, the difference between the first and second terms is 

 13 ; which represents from the 19th to the 20th June, the meas- 

 ured change in declination ascending. Between the 21st and 

 22d June, the difference of the third and fourth term is 8 ; but 

 the change is to declination descending. Between the 20th and 

 21st, the difference is only 2; and although it might by possibil- 

 ity be that the ascending declination (or descending series) was 

 continued on to the 2 1st June, and that the* turning point took 

 place there or farther on, we are restrained from such an assump- 

 tion from the large difference between the third and fourth terms, 

 too great to consist with an incipient reversed motion. The crisis 

 then must have occurred, as we have all along supposed, between 

 the second and third term ; and although this third term is yet less 

 than the second, it bears in fact a contrary sign. The descend- 



