Terrestrial Magnetic Intensity. 63 



There is another chronometric correction worth mentioning, 

 arising from the necessarily imperfect division of the seconds' 

 circle of an enamelled dial-plate. In my watch this amounts 

 to a sensible quantity, and has often given an apparent discre- 

 pancy ]Xo the partial results of a series for which I was not 

 prepared. Upon investigation, I find, however, that the effect 

 upon the mean will always be so insignificant as to be hardly 

 worth notice. 



13. II. Arc. — A correction due to the motion of the mag- 

 netic pendulum in circular arcs, cannot be considered as a 

 constant quantity, and therefore not affecting relative results, 

 1. Because the rate of diminution of arc varies considerably 

 in different experiments, and is directly deduced from the ob- 

 served law of diminution of arc; and 2. Because we some- 

 times have to compare observations of 100 vibrations having 

 an initial semi-arc of 10°, with 300 vibrations beginning at 20°. 

 The latter case having alone been considered by Hansteen, 

 I reinvestigated the theory of the correction, and confirmed 

 his numbers. 



14. It is assumed that the arc diminishes geometrically in 

 consequence of resistance, the time increasing arithmetically. 

 The best observations I have made confirm the truth of this ge- 

 neral admission. Again, we have to recollect that, in conse- 

 quence of the degradation of the arcs, the reduction to infinitely 

 small arcs for the vibrations between the 0th and the 300th, 

 will be greater than between the 10th and 310th, &c, and that 

 the mean of all the corrections (taking this variation into ac- 

 count) must be applied. The law of the diminution of arc, or 

 the factor representing the ratio of the arc of one vibration to 

 that of the immediately preceding one, will be at once deduced 

 from observing after how many vibrations the arc is halved. 

 Let m be that number, then if r be the factor in question, 

 r™ = \ ; whence r =™ \/ £, which is known ; and therefore m 9 

 together with the initial semi-arc of vibration, may be used as 

 the arguments for entering the following table of correc- 

 tions * : — 



* Investigation. — Let » be the initial semi- arc of vibration (taken in parts 

 of radius), and r the ratio of its diminution by resistance in a single vibra- 

 tion. 



Then, for the 1st, 2nd, 3rd, 4th, wth vibration, 



The arcs will be ct, et r, » r 3 , m r 2 , u r 



And, by mechanics (Poisson, art. 184.), the times occupied by these vibra- 

 tions will be (the time of an infinitely small vibration being unity), 



1+ T6' 1+ T6-> 1+ -W 1 +T6-- 1 + f T 



And the mean duration of the n vibrations is, 



