( 452 ) 



For the proportion of the vis viva of the reduced system to that of 

 the pendidiim referred to we can write ^) : 



M u" : m, a,' ^,' = M' («/'"V : m, a,' y.,' = 



= 31' («'^"V ih - ^-y ' rn, a,^ (§,^'" ^ = (/, - Xf : c,U,\ . (21) 

 If now Cj is not small, as in case B, then we have in this manner 

 already proved what was put. In case A we substitute P. = /^ — rf 

 in the cubic equation (14) after which we lind easily at first appro- 

 ximation, Cj being hkewise small-), ö=I^ — ).=c^''I'I^:{I' — /j, by which 

 what Avas put is likewise proved. 



In case C finally, which occupies our attention at present, ensues 

 from (20) for the rapid principal oscillation 4 — ?. = (c^'-j-Cj^) I'l^ : (/' — /j ; 

 from which is evident after substitution of ^ and c^ for /^ and c^ in 

 (21) the correctness of the theorem also for this principal oscillation, 

 hence a fortiori for the intermediate one; unless Cj be small but yet 

 much larger than c^, which restriction does not exist for the inter- 

 mediate principal oscillation. 



15. From these results must be inferred that in the case C under 

 consideration the rapid principal oscillation as well as the intermediate 

 one when once set in motion will each be able to maintain them- 

 selves under the influence of the motive works, when the condi- 

 tions of friction in the frame are not too unfavourable. However, the 

 intermediate principal oscillation will have, if the difference in rate 

 between the two clocks "was originally very slight, a considerable 

 advantage on the rapid one, the motion of the frame being much 

 slighter still in the former case than in the latter. And this will 

 probably be the reason that in the experiments of Huygens as well 

 as in the later ones of Ellicott evidently the intermediate principal 

 oscillation exclusi\'ely ') or at least chiefly ^) presented itself. 



1) According to (10), (15) and (16) taking at the same time note of the sig- 

 nification of 1 1, «1 and ki. 



') For q small and Co not, the proof runs in the same way, although the 

 expression for I becomes a little less simple. 



"-') With Huygens. In his experiments the masses of the pendulums were certainly 

 slight with respect to those of the frame, so that without doubt Ci and Co were 

 small and the case C was present. 



^) With Elugott, where at least at first according to the observed transferences 

 of energy also the rapid principal oscillation must have been present. Although 

 Ellicott used according to his statement very heavy pendulums, we have probably 

 also the case C with him. If we do not assume this then it is more difficult 

 still to make the perfectly equal rate of his clocks tally with the observed trans- 

 ferences of energy. The presence of two principal oscillations evident from these 

 would have been continued indefinitely in case B, so the clocks would have 

 retained an unequal rate. 



