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LVI. On the Motion of a Pendulum affected by the Earth* s Ro- 

 tation. By Septimus Tebay, Mathematical Master, Bruce^s 

 Academy, Newcastle-upon-Tyne. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



SHOULD you deem the following brief solution of this in- 

 teresting problem worthy of notice, its insertion in an 

 early Number of your valuable periodical will much oblige 



Your obedient Servant, 



Septimus Tebay. 



Let the centre of the earth be the origin, and its axis the 

 axis of X, and at the commencement of the motion let the plane 

 scz coincide with the meridian. Let r be the radius of the earth, 

 I the length of the pendulum, /? the angular velocity of the earth, 

 6 the inclination of the plane of the pendulum to the plane of 

 the meridian, e, e\ e" the directing angles of the vertical line, 

 xyz the coordinates of the centre of oscillation, p the line from 

 this point to the origin, 77, 77', 77" the directing angles of p, and 

 ^ the inclination of p to the vertical. 



The dynamical conditions of the problem are represented by 

 the equation 



(J^ + Gcos6)8^+(J + Gcos6')ay + (J+Gcos8'')s^=0(l.) 



(Poisson, Traite de Mecanique, No. 531), G being the whole 

 attraction of the earth, supposed at rest, on a particle at its 

 surface. 



By the geometry we have 



a!=p COST), y=p COST)', z=zp cos rf". 

 And € being equal to the colatitude of the place, 



cos e' = sin € sin fit, cos ^' = sin e cos ^t. 

 Let €+</) be the polar distance of the ball of the pendulum, and 

 yjr its longitude measured from the meridian of the place, ^ and 

 •^ being necessarily small. We have 

 cos 7) = COS (e + <t>), 

 cos 7)' = sin (e -f (j)) sin {0t -\- ylr), 

 COS 7]l'=siD. (e + c/)) COS {j^t + '^). 

 Also^ putting r + /= R, we have 



R2+p2_2Rpcosx=/^. 

 Hence, as far as small quantities of the second order. 



