394 Astronomical and Nautical Collections. 



.00029388 A 2/= v-+ (2.1312 - .5s')-l+ 2417.5 v^^ 

 s s'^ s' 



+ (2576 - 606 s"- + 4 113400 ?;^ ) — 



s* 



+ (12 273 040 -2 468 000s'+ 10 570 000 i.^)u if 

 + ( [2 045 500 - 41 1 333 s' + 5 260 000 r^ . (4.2624 — s«) 



+ (8 182 000 - 1 234 000 ss + 7 013 200 v^) 3403 v^)^ 



s^ 

 + . , . 



In order to compute the horizontal refraction in two portions 



from this formula, supposing its magnitude to be about .01, we 



may begin by taking half of this quantity, and make r = .005, 



- and r^ = .000025, s being = 1 , and v — 0; we shall then have 



p A7J ~ .00004578 + .000001231 + .000000082 + . . . 



Now an important question is here, to be considered ; whether 

 we are to content ourselves with simply adding together these 

 terms as constituting the whole series, or whether it is justifiable 

 to assume something more from analogy, or from probability, 

 upon general principles, excluding all arbitrary conjectures 

 depending on private reasons or imaginations. 



If we admit the propriety of interpolating a table by the me- 

 thod of differences, which has seldom been called in question, it 

 can hardly be denied that, with proper caution, the method of 

 differences may very safely be applied to the continuation of a 

 converging series. Supposing the progression to be very nearly 

 geometrical, it cannot be questioned that there is a probability, 

 approaching practically to a certainty, that we shall be much 

 more correct if we suppose it to continue strictly as a geome- 

 trical progression, than if we break it off abruptly : and when the 

 difference from a geometrical progression is more considerable, 

 it seems at first sight natural to take the ratios or logarithms of 

 the quotients, and to continue them by means of the differences. 

 But where, as in the present instance, there are only three terms 

 computed, and we have only one difference of these logarithms, 

 the continual employment of this difference must be erroneous, 

 because it will always ultimately lead us to a diverging series, 

 affording the succeeding terms too large in proportion to the 



