( 447 ) 



^'S. 3. tlie opposite is the case. For the slow prin- 



cipal oscillation the oscillations of both pen- 

 chihims are either of the same order as those 

 of the frame or smaller still ; the latter is 

 the case when the third cause mentioned in 

 4: rapid principal osc. note 2 of page 446 is at work. 



rapid pendulum L Suppose now jt', Ji, and jt, to be small oscilla- 

 tions belonging respectively to each of the 

 three types of the principal oscillations, namely 

 the slow one, the intermediate one and the rapid 

 one, each having the same small quantity 

 ;c inlerm. principal osc. of total energy 6= J'-j- V; then every 

 <> slow pendulum li compound oscillation can be represented by 

 a}^K':n:'-'rK^n^-^K^3t^ and its total energy will 

 be equal to {IC' + K,-" + K,-") e. 



Let us then start from an arbitrary com- 

 pound oscillation for which K\ K^ and K^ 



4 reduced system have moderate and mutually comparable 



:irslow principal osc. ^^^^^^ , -^ -^ ^j^^^ ^^^^^, ^^^^ ^j^^ ^^^^.^^^ ^^ 



one clock, namely the one with the rapid 

 pendulum will be dependent almost exclusively on the rapid prin- 

 cipal oscillation, that of the other clock on the intermediate one. 

 It is true, that slight periodical deviations in the amplitudes will 

 present themselves, which are due to the two other principal oscil- 

 lations, but these can have no influence of any importance on the 

 periods according to which the motive works regulate their action; 

 so that therefore one of the motive works will be able to contribute 

 to the sustenance of the motion K^n^, the other to the motion K^jt,^ 

 but neither of them to the sustenance of the motion K'jx'. So this 

 will vanish first. 



What takes place furthermore will depend on the power of the 

 motive works, and on the frictions presenting themselves during 

 the motion of the frame. If those powers are great enough to 

 conquer the frictions when the pendulums deviate sufficiently to keep 

 the motive works in movement, a motion K^ jt^ -\- K^ ji^ will remain, 

 where the values of /^ and K^, thus also of their proportion, will finally 

 depend exclusively on the power of those motive works and on the 

 frictions. A theorem the proof of which we shall put off to § 14, to 

 be able to give it at once for all cases, shows that in general such 

 a motion can be sustained rather easily ; it is the theorem that for 

 principal oscillations whose A differs but slightly from Zj or 4 whatever 

 may be the cause, the kinetic energy of the motion of the frame 



