162 Intelligence and Miscellaneous Articles. 



The variation of the element r produces a change in the duration 

 of the oscillation ; but this small change did not seem to us capable 

 of being determined with certainty by experiment, as the least change 

 of temperature in a very long pendulum produces a much greater 

 change in the duration of the oscillation. The only appreciable 

 effect is the variation of the small axis. 



The secular part of the perturbing function, making 5=0, is 

 represented by the formula 



R _ / y ,i—r i.3.5...(2t-l) / fl 2 V 



""" V^+^ i=0 L 2.4.6... 2i \a 2 + hy X 



*"=« * 2.4... (2i-4?) \a~) _T 



where f denotes the action of the perturbing force on the unit of 

 mass at the unit of distance, the angle a being reckoned from the 



right line O A. In this sum the whole number i varies from to „ 



i— 1 

 or to — — , according as i is even or odd, and the number i from to oo ; 



C|' denotes the number of combinations of i objects taken i' and i'. 

 There is hence obtained 



db a 2 n . 



i-r i.3..(2i-i) / a 2 V 



X - 2 L 2.4...2i \a* + h*) X 



i>=lt * 2.4...(2i— 4i') \ d z ) \' 



The latter formula shows that the pendulum, which at first described 

 a plane oscillation, afterwards describes an ellipse, the smaller axis of 

 which increases proportionally to the time, and in a direction such that 

 the velocity at the summit of the major axis is directed towards the 

 point from which the perturbing force emanates if it is attractive, 

 and in the opposite direction if it is repulsive. 



The celebrated experiment of M. Foucault has shown that the 

 plane of oscillation of a pendulum appears to be displaced in conse- 

 quence of the motion of the earth. Let us suppose that the terminal 

 ball is magnetic, and that on the two sides of the plane and at 

 the two extremities of the oscillation magnets are placed ; it will 

 soon be observed that the pendulum describes an ellipse the major 

 axis of which is displaced as in the experiment we have mentioned, 

 and the minor axis of which increases proportionally to the time. If 

 the positions of the two magnets be reversed, the small axis di- 

 minishes progressively, is annulled after the lapse of a time equal 

 to that which had taken to form it, after which a new ellipse begins 

 to form in the opposite direction. If the magnets are powerful and 

 near, the increase of the small axis is seen at once ; in proportion as 



