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XLVII. A Mathematical Theory of M. Foucault's Pendulum 

 Experiment. By the Rev. J. Challis, M.A.,F.R.S., F.R.A.S., 

 Plumian Professor of Astronomy and, Experimental Philosophy 

 in the University of Cambridge*. 



THE remarkable experiment which recently attracted the 

 attention of mathematicians, as being a practical demon- 

 stration of the earth's rotation, has already received various illus- 

 trations and theoretical explanations ; but I am not aware that 

 any explanation has yet been derived, according to rule, from the 

 differential equations of motion, which on the principles of dy- 

 namics especially belong to the problem. I propose in this com- 

 munication to form those equations, and by means of them to 

 show that the facts observed may be explained on the hypothesis 

 of the earth's rotation. 



The problem may be generally stated thus : — To determine 

 the motion of a ball suspended from a given point of the earth 

 by a slender cord, and acted upon by gravity, the earth being 

 supposed to revolve about an axis with a given angular velocity. 



Conceive a line to be drawn through the point of suspension 

 of the ball parallel to the axis of rotation of the earth, and a 

 motion equal and opposite to that which this line has in space 

 at any instant, to be impressed on all particles of the earth inclu- 

 sive of the cord and ball. The line will thus be brought to rest, 

 and all other points will begin to move as if they were revolving 

 about it with the earth's angular motion. By supposing this 

 axis to remain at rest, and the angular motion to continue, the 

 motions we are about to consider will not be relatively altered. 

 On this supposition, the direction of the force of gravity, bein°- 

 always perpendicular to the earth's surface, will revolve about 

 the same axis. Consequently, the problem above enunciated is 

 identical in its dynamical conditions with the following : — 



To determine the motion of a ball suspended by a slender 

 cord from a point in a fixed axis, and acted upon by a con- 

 stant force in the direction of a line making a given angle with 

 the axis and revolving about it with a given angular velocity. 



Conceive to be the point of suspension, and OX, OY, OZ 

 to be fixed rectangular axes, of which OZ (drawn downwards) 

 coincides with the axis of rotation. OA is the direction of gra- 

 vity, making a constant angle AOZ(X) with OZ, viz. the co-lati- 

 tude of the place where the experiment is made. P is the posi- 

 tion of the centre of the ball, OP = « the length of the cord, and 

 x, y, z are the coordinates of P at the time /. Let co = the 

 earth's velocity of rotation, and consequently the angular velocity 



* Communicated h\ the Author. 



