AS EXHIBITING THE ROTATION OF THE EARTH. 19 



the plane of yz of tlie circular segment included between the arc and chord of 

 vibration), multiplied by n cos ?.. This multiplier, which is the component of the 

 earth's rotation about a diameter of the earth normal to that passing through the 



locality, indicates that the term 2)i cos T^j y dz expresses a disturbance produced by 



this complementary component. As this term for vibratory motion is small and 

 periodic, passing through nearly equal positive and negative values in the course 

 of a double vibration, it follows that u sin "X, expresses the mean increment of angu- 

 mortion, or, in other words, that, to the plane or spherical vibrations exhibited by 

 the pendulum over a motionless earth, there is, superadded, in consequence of this 

 rotation, a uniform azimuthal motion measured by the earth's rotating velocity mul- 

 tiplied by the sine of the latitude. This is the material fact or peculiar feature of 

 the freely suspended pendulum, and we see that it is exhibited by equation (4) 

 generally for all ordinary vibrations, whether plane or spherical. We shall see 



hereafter, however, that the disturbing term of equation (4) '2n cos ;i J y dz expresses 



a tendency to a like motion about the complementary axis, and that, on tlie sup- 

 position of an infinite velocity, this tendency may be realized, and the plane of 

 motion, by the joint effect of the two components, turn around a parallel to the 

 earth's axis, with an angular velocity equal and contrary in direction to n. 



In the case of very small deviation^ from the vertical, the equations (3) may be 

 solved as follows : The variations of z then become of the second order of minute- 

 ness compared with those of cc and ;/< and oniitting them we have between x and y 

 and their differentials the relations 



d'^x , Nx . dii 



^^^ -^+-^-2«s.n?-/=0 



d^y Ny ■ dx 



-^+-^+2»i sin ^^^ =0 



the integrals of which are (Gregory Examp. p. 390), 



a:=-f 2 {D cos ^t-^E sin (it) 

 ^ 3^=— 2 {D sin (it—E cos [it) 



in which to /3 is given both the values obtained from the quadratic equation 



ffo — ttj /3"=fli /3 * 



in which «o^ ? '^i^= — 2u sin 7., a^^\ ; hence solving the quadratic 



f>-\l±U^i{^) 



Substituting the values of Oo, &c., and oniitting the second term under the radi- 

 cal as inappreciably small, we have 



g_ 

 I 



y=G sin {fit — t) in tlie given equations. 



* This equation, i- j3'=2)i sinji/3, can be got by substituting the integrals x^Ccos (9/ — t), 



