22 /. H . Alexander on a New Formula for Interpolations. 



The epochs of the ordinates for interpolation are then 125 ; 1*50; 

 1*75, respectively. I need only give, for illustration, the substi- 

 tution of one of these epochs in the formula III ; which as there 

 are four terms is the one properly applicable. Taking, then, the 

 value of n at lo, and substituting for a, 6, c, d, the numerical 

 distances in their order, we have 



I 



0-5. -0-5. -1-5 



+ 



1 . 2.3 . 

 0-5.- 1-5.-2-5 



. 260290 



0-5, -05, -2-5 

 1.2.1. 



. 237178 



1.2.1. 



. 2 1 4427 



0-5.-1 



1.2.3. 



O.K 



—-.192016; 



00625 . 260290 -0 3125 . 237178 + 09375 . 214427 



+ 0-3125 . 192016; 



16268"125 - 74118 // -125 +201025-3125+60005 ; 

 203180-3125 = 56° 26' 20 / '-31 



A similar substitution for the other epochs will enable us to trans- 

 form the series, as under : 



Nov. 1st — Noon. 



Distances, 

 do. Wallace, 



53 C 20'JG" 



II 



3 p. M. 





6 P. M. 



9 P. M. 



54°53'7"9G 

 r>4°53'8" 



56 D 26'20 / '*31 



56°26'20" 



| Midni 



iht 



57 C 6!>'53''-20[51KJ3 , 47 



57°59'53" 



i 



« 



It is thus seen that the formula, with but little (if any) more 

 figure-work, is not deficient either in accuracy or in precision. 

 The arithmetical operations might even have been abbreviated, 

 by dividing at once all the terms in the series of distances by the 

 first term. The answers would have been produced by multi- 

 plying at least each result by the same constant factor. But 

 artifices of this kind belong, of course, to the working of any 

 formula. 



Such a division, however, might have been useful for another 

 purpose, as I have already mentioned ; namely, to determine the 

 order of the equation to be employed. As in the present instance, 

 the object was only a comparison of methods, and as all four 

 terms had concurred in the method of Wallace, they were of 

 course used equally and without question in my own. It is how- 

 ever easy to see that many practical cases may occur, in which 

 more terms are given than would be necessary, or even proper, 

 to employ under the physical relation of the question. In the 

 example just now from Weisbach, there were four terms given 

 too, but only three were employed. In the last example, the 

 terms were no more than necessary, as will be seen by the follow- 

 ing transformations and parallelisms. 



Epochs in time. 



I. 



2. 



3. 



4. 



&c. 



Distances divided by 1st terra. 



1. 



1-116 



1235 



1 -3. r >6 



&.C. 



-V epochs : 



1. 



1414 



1-739 



2 — 



&c. 



y/ epochs : 



1. 



1260 



1-442 



1-587 



&c. 



%/ epoch*: 



1. 



1184 



1-316 



1-414 



<frc. 



