Considerations on the Problem of Latitude Variation. X7 



the so-called ''dead motion" of the screw by turning it always in 

 the same direction. The same principle is applied in determining 

 the value of a division of the spirit level, always making the bubble 

 come to rest from the same direction. This precaution implicitly 

 recognizes the existence of a failure in the function of the level. 

 It seems to me rather curious that in the practical work of star 

 observation the level is taken as a perfect instrument and the pre- 

 caution needed in the above case is utterly neglected. Here I 

 intend iu the following lines to show that there actually exists such 

 a failure in the function of the spirit level. I will first proceed 

 from the theoretical standpoint and then give an experimental 

 proof. 



The motion of a level buljble can be looked upon as analogous 

 to a damped oscillation of a simple pendulum ; the equation of 

 motion accordingly takes the following form, — 



d'd ^ dd „^ _ 



4- 2p — - + n-d = 



df "■ dt 



where/) and n are constants and 6 is the deviation of the bubble 

 from the position of equilibrium. The applicability of this equation 

 was experimentally tested by Bonsdorff'^ and further discussed by 

 Orloff.') 



The integral of this differential equation comes out in the fol- 

 lowing three forms, according to the cases, n'>'j), n-'p, yi<p respec- 

 tively, — 



( i ) nz>p, d = e~"' [cii cos (it + a.sln fd] 



(ii) n=p, 6=6-"' [a,t + a,] 



(iii) n'<p, 6 = b, f;-V + 6,e-V ip' = iv—p- 



whereJrta = pVi-Jl-'j^. | 



Here the quantity n depends on the radius of curvature of the level, 

 and the quantity p is a function of the bublde length and of the 

 viscosity of the liquid. 



1) Mitteilungen der Nikolai-Hauptsternwarte zu Pulkowo, Bd. II, p. 43. 



2) Ditto Bd. II, p. 137. 



