16 T II E P E N D U L U M A X D G Y R S C P E 



da- 



(2) Y= 2n ^sin^ 



dx 



Z 2n -,- cos a. 

 at 



In which /I is the latitude and n the 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 others referring to the moving axes which are used in this 

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

 the following considerations. 



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



given latitude will be - _ ' S (r being the earth's radius). If it has a small 



/ , , 



dx \ nrcos *+ 



relative velocity --,- to the cast, the centrifugal force will become - 



dt dx\ r cos * 



Subtracting the former expression from the latter (omitting , 2 j we get for the 



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



U>t 7 (.it 



UX 



The component of this in the direction of the axis of y will be 2n sin ?. -, , which 



corresponds to the value of Y (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 



~dx 



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 cast 



avec toute la certitude que les sciences physiques comportent, cependant une preuve directe de ce 

 phenomena doit intcresser les geomotres 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 affected, 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 aucnne influence appreciable sur son mouvement." (Journal de I'Ecole 

 Poly. Cahier, 26, p. 24.) It is true, 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 cumulative in changing the 

 azimuth of the freely suspended pendulum. 



These same disturbing forces, introduced along with the attractions of the sun, moon, and earth, 

 into the general equations of equilibrium of fluids, produce in a very simple manner the differential 

 equations for the tidal motions. ( Vide American Journal of Science, 1800.) 



