1 I ; r E N D U L U -M A^V GYROSCOPE 



16 



dz 



ft Of 



(2) i'=-2«e^«"^^ 



Z =—2)i -, ■ COS ;i. 



111 wliicli >. is tlio latitude and n tlie angular velocity of rotation of the earth. 



These analytical expressions make their appearance in the transformations of 

 the equations of motion, near the earth's surface, expressed in co-ordinates referring 

 to fixed axes, into otliers referring to tlie moving axes ^vhic]l are used in this 

 analysis. But these forces can be obtained and their origin bettqr understood by 

 tlie following considerations. 



The centrifugal force of a material point at rest on the earth's surface at the 



2 2 2 1 



given latitude will be ^' ^ ^^^ {r being the earth's radius). If it has a small 



rcos;i / ^^2. 



(«rcos;i+-^j 



dx ^ dt 

 relative velocity jj to the east, the centrifugal force will become . 



(Xt ^2 ^ COS A) 



Subtracting the former expression from the latter (omitting -,^, j we get for the 



centrifugal force arising from the relative velocity - - the expression 2« -. 



dt J dt 



dx 



The component of this in the direction of the axis of ^ will be — 2n sin X , , which 



corresponds to the value of F (equations 2) of Poisson, and is to be added to the 

 second member of the second of equations (1). 



The component of the force just calculated in the direction of the axis of Z is 



— 2n cos /I — 

 dt 



This force corresponding to Poisson's value of Z is to be added to the second 

 member of the third of equations (1). 



A body moving on a meridian of the earth's surface from south to north will 

 have the moment of its quantity of motion, with reference to the earth's axis, 

 diminished ; in virtue of which it will press with a certain force towards the east 



avec toute la certitude que les sciences pbysiques coniportent, cependant une preuve directe de ce 

 phcnomune doit intcrcsser les gcometres et les astronomes." The former, in making a partial appli- 

 cation to the pendulum, of his investigations, absorbed, apparently, in the single object of proving 

 that the accuracy of the instrument as a measure of time was not afiTected, has inadvertently assumed 

 that the disturbing force normal to the plane of oscillation is "trop petite pour ^carter sensiblement 

 le pendule de son plan et avoir aucune influence appreciable snr son mouvement." (.Journal de I'Ecole 

 Poly. Cahier, 26, p. 24.) It is truo, indeed, that the force he mentions, even if permitted free 

 action, will have but an inappreciable influence upon the time, and none whatever when, as in the 

 chronometer, the plane is constrained to fixedness ; but the effect is cumulatwe in changing the 

 azimuth of the freely suspended pendulum. 



These same disturbing forces, introduce.! ab.nir will, the attractions of the sun, moon, and earth, 

 into the general equations of equilibrium „r fhmis, produce in a very simple manner the differential 

 equations fur the tidal motions. {Vide American .Journal of Science, ISGO.) 



