24 J. H, Alexander on a New Formula for Interpolations* 



The numerical transformations of these, result in three main 

 equations, viz. 



(«). 17476-84*/ -216 c+189 6; 



(/?). 50380 = 220 d- 594c + 540 A; 



( r > 86858 = 364 d - 1001 c + 924 b ; 



Multiplying («) by 4£; and subtracting from (y), we eliminate rf J 



and leave (5) 1 1 128| = 105 6-65 c. 



Again multiplying («) by the fraction f f ; subtracting from (£)j 

 and multiplying the remainder by 2£, we obtain 



(•> 10755| = 105 6 -66 c; 

 and subtracting this last from (<?), we eliminate b ; and deduce 



c = 372|. 

 By similar substitutions, 6 = 336^|| ; and 



With these values we can now interpolate any term. For the 



one in question we have n = 9 ; andl= 8 ' 7 ' 6 . 409j» T - 8 ' 7 ' 5 372§ 



2.3 3 2 



+ 9 • 336^|| - ' o . 301 ; whence the distance required, 



or the 9th term, (I) = 586 fa or 586-295. The result of Wallace, 

 586*3 may be said to be absolutely identical. 



4. I shall give but one more example of this kind. In an experi- 

 ment of Coulomb, it was observed that with ropes of 12*5, 20, 

 and 28 lines in circumference, passing over a pulley of four inches 

 diameter and strained with a constant weight, the rigidities were 

 equivalent to 11 ; 27; and 50 pounds, respectively: what will be 

 the equivalent rigidity, other things being the same, of a rope 16 

 lines in circumference ? 



Here we have the two series, corresponding as under : 



In. In. In. 



Circumferences: ' 12-3. 20. 28. 



do. divided by 1st term : 1. 1-6 2-24 



Rigidities: 111b. 27 lb. 501b. 



In this case, the value of a (formula II.) is known ; the value of 

 b and c are obtainable from the equations of 271b. and 501b. to the 

 corresponding values of n ; viz. 1*6 and 2-24 respectively. These 

 being determined they are substituted in the same formula, taking 

 also the value of n at 1-28 ( T W)> l ^ e epoch of the required or- 

 dinate. The processes themselves are so purely artificial and so 

 similar to what was given in the last example, that they need not 

 be exhibited here. It is enough to say that they result in making 



Z= 1777 lb. 



Weisbach's method has given the same value of this term at 

 20-39 lb. ; which is manifestly erroneous and would accuse a con- 

 vergence of a series which is plainly diverging. It could not by 

 possibility be so much, even were the interpolated circumference 



* 



