334 On M. Foucault's Pendulum Experiments. 



suppose the ball to have initially no velocity relatively to the 

 plane AOZ, and also to be started in a direction perpendicular 

 to the meridian at a small distance from the vertical. Thus we 

 shall have in equation (a), Vj = 0, l =\ nearly, and the motion 

 will at all times be small. The last term of that equation will 

 be insignificant, both because or is very small, and because 

 sin 2 6— sin 2 \ will have in each excursion small positive, and 

 small negative values of nearly equal magnitude. Thus very 

 approximately, 



V* = 2 ff (z'-h) (7) 



. . . dz* 



Again, in the equation (/3), the term involving -^- will be in- 

 significant, the motion being by supposition nearly perpendicular 

 to the axis of s'. Also the factor a> 2 makes the last term very 

 small. It is, however, to be observed, that while the product 

 y'z' is alternately positive and negative in each excursion of the 

 ball by reason of the change of sign of?/', the product x'y' changes 

 sign slowly, the excursions being nearly in a plane. The effect 

 of the term involving x'y' might possibly be sensible in a delicate 

 experiment. Neglecting the small terms, we have 



y'^. _^ = H-a)COsX(*' 2 + y 2 ). . . (8) 



The equations (7) and (8) apply to any small vibrations 

 affected by the earth's rotation. If the term in (8) involving &> 

 be omitted, the equations are the same that would be obtained 

 on the supposition of no rotation, in which case, as is known, 

 the ball moves in a small ellipse, the axes of which pass through 

 the vertical and have given inclinations to the plane of the meri- 

 dian. The effect of the rotation of the earth, from what has 

 been shown, is to cause the ellipse to revolve relatively to the 

 plane of the meridian about the vertical from left to right with 

 the angular velocity to cos X. 



The case of larger excursions might be treated in the manner 

 employed by Mr. Airy in the Memoirs of the Royal Astrono- 

 mical Society, vol. xx. p. 121, the foregoing result being taken 

 as the first approximation, instead of supposing tbe axes of the 

 ellipse to have fixed directions with respect to the plane of the 

 meridian. 



Cambridge Observatory, 

 April 5, 1852. 



