172 OBSERVATIONS OF THE TERRESTRIAL 



astronomical observations. In the present series the rate was 

 observed at the commencement and end of each group of observa- 

 tions by the former and easier method. The amount of the 

 correction due to rate is in most cases very small, the correction in 

 the time of 100 vibrations corresponding to a daily rate of 2 s 

 being less than O s> 01 with the slowest of the needles employed. 



2. Professor Hansteen has applied a correction for the arc of 

 vibration, so as to reduce the time to that corresponding to in- 

 finitely small arcs. The correction is investigated on the same 

 principles as that usually applied to pendulum observations. It is 

 however more complicated in its form ; for, instead of a single 

 series of vibrations (as in the case of the pendulum), we have here 

 seven distinct series, each commencing from a different arc. The 

 principle, however, seems hardly applicable in the present instance. 

 It is assumed that the successive arcs of vibration decrease in 

 geometric progression, as they must necessarily do if the resistance 

 of the air be proportional to the velocity. This is found to hold 

 good in the vibrations of the pendulum when the arcs are very 

 small ; but it is by no means true when they are so considerable 

 as those in which the horizontal magnetic pendulum is made to 

 vibrate. Where, however, the vibrations commence from the same 

 arc, and the terminal arc does not much vary, the correction itself 

 may perhaps be disregarded. In the following observations, in 

 which the initial arc was 20, the 360th or terminal arc was 

 generally 2, and was in all cases included between the limits 1 

 and 4. In such cases, then, the correction must be, nearly, a 

 constant quantity ; its application to the observed times is there- 

 fore nearly equivalent to their multiplication by a constant co- 

 efficient, and the ratio of the times (with which alone we are 

 concerned in this class of observations) remains unaltered. For 

 these reasons no attempt has been made to introduce a correction 

 for the arcs in the following results ; but the terminal arcs are 

 given, so as to put the reader in possession of all the circumstances 

 of the observation. 



3. By far the most important correction is that due to tem- 

 perature. If T' be the observed time of 100 vibrations correspond- 

 ing to the actual temperature t', and T the corrected time cor- 

 responding to the standard temperature t, the correction is 



T- T' = T' t-f;. 



