

/. H. Alexander on a New Formula for Interpolations. 25 



16-25 inches (the mean between 12-5 and 20), and the differences 

 of the rigidities constant in the first place. I have not, however, 

 farther investigated the causes of his error. 



5. I shall terminate these numerical illustrations with one more 

 instance; in which both coordinates undergo interpolation, simul- 

 taneously. This, as it is one of the most complicated cases, is 

 also one of the most interesting applications of the formula which 

 developes itself here with remarkable simplicity. The instance 

 is the solstitial observation of Gassendi, in 1636, at Marseilles, of 

 the shadow of a lofty gnomon. Supposing the gnomon divided 

 into 89428 parts, he found the length of the shadow at noon on 

 the respective days, as under : 



Epochs. June 19. June 20. June 21. June 22. 



Corresponding values of n. 1. 2. 3. 4. 



Lengths of shadow: • 31766 31753 31751 31759 



The question is to ascertain by interpolation at what moment the 

 shadow was the shortest ; which is also the moment of the 

 solstice. 



It has been usual in questions similar to this, in order to avoid 

 the resolution of equations of a degree higher than the second, 

 not to involve all the terms at once ; but to group them into sets, 

 the mean of whose several results may be taken as the absolute 

 °ne. It need hardly be said that such grouping and permutation 

 lead only to partial solutions, which however approximate, do not 

 exhaust at once all the elements given. The present method 

 allows of the simultaneous employment of all these : but as there 

 are only materials for one equation, while there are two unknown 

 quantities, resort must afterwards be had to the higher analysis to 

 determine when the shadow-length and the corresponding ordi- 

 nate becomes a minimum. 



Taking, then, our formula III, and leaving n in that, as un- 

 known : and substituting for a, 6. c, and t/, respectively, the 

 values of the shadow-lengths in the order given above ; and exe- 

 cuting the necessary numerical transformations, we obtain the 

 sum of the terms which, when the unknown quantity is a mini- 

 mum, must be equal to zero ; as under, 



in* + ten 2 - 31£ *i + 3I791 = 0. 



Changing the signs, and differentiating in the ordinary methods, 

 this leads to an equation of the second degree, viz. 



whose resolution gives a positive value of rc~2*69; which is to 

 be counted in the direct order from 18th June, and corresponds of 

 course with June 20 d , l(> h , 33 in -6. as the moment of the solstice. 

 It is evident that the order in which the terms should be taken 

 W a case of this kind, is quite arbitrary and ought to be immate- 

 rial to the result. Accordingly, if we invert the present order and 



Second Series, Vol. VII, No. 10.— Jan., 1349. 4 



