24 



THE PENDULUM AND GYROSCOPE 



If the amplitude's of the vibration arc very minute, so that may be substituted 

 for sin m (1), the integral becomes 



.,)N ■ ^^"^^,j sin ;i— ^w cos ?. cos^) 



^' ' dl o 



expressing the precession in azimuth, n sin ;i, precisely as it results from the former 

 analysis (equation 6). The sliglit periodic disturbance expressed by the second 

 term of (i)) escaped that analysis, however, owing to the omission of the terms 



involving dz. 



In the above integrals the angle ^ must be taken at 180° greater or less on one 



side of the vertical than on the other, and the parts of ^-|* expressed by it will have 



contrary si"-ns. Hence the curve described in each complete excursion, disregard- 

 in'' the superadded uniform azimuthal motion expressed by the first term of the 

 second member of (9), will have the form of an excessively attenuated leminiscate, 

 or figure of 8. " 



For greater amplitudes equations (8) will apply until becomes nearly equal to 

 180°; if Q equals or exceeds 180°, it cannot he assumed that the pendulum will 

 pass through the zenith (the condition for ^ to remain nearly constant), and the 

 integral becomes inapplicable and erroneous. 



The foregoing integrals involve the condition that the pendulum shall pass 

 through the vertical, and imply that vibration is induced by propulsion from a 

 state of rest in the vertical. But, in the usual form of the experiment for exhibit- 

 ing the rotation of the earth, the pendulum starts from a state of relative rest at 

 the extremity of the initial vibratory arc. 



If we disregard the symbolic integral of (7), as may be done, since the minute 

 periodic disturbance it measures has no intiuence upon the permanent azimuthal 

 motion, that equation will become 



f \ d(p . ^ , C 



dt ^l' sin^ d 



The second term of the second member is identically the equation of the " spheri- 

 cal pendulum." The latter, we know, exhibits an azimuthal motion of the apsides 

 of its orbit, very minute when C is small, but incomparably greater than the horary 

 azimuthal motion when, C being large, the conjugate dimensions of the orbit 

 approaches equality to the transverse, and of which the limit corresponding to per- 

 fect equality of these dimensions is for one vibration, 



00. 7tr-^^=J_-i 1 



as may be deduced from the expression for U„, par. 731, Peirce, Analyt. Mech., or 

 from expression [79] and [83] of Mec. Cel. (Bowditch), by making a=.b and deter- 

 mming the corresponding values of c and dt. 



In the case under consideration the value of C will be determined by making 

 dl ~^ ^°^ ^^^^ commencement of motion, 0^,9.,, hence 

 C=—n V sin X sin- 0, 



Sec Additional Notes, p. 51. 



