( 449 ) 



itself in both clocks '). In the second case the clockwork stops 

 entirelj ; the corresponding principal oscillation vanishes, and the 

 pendulnm performs onlj passively the slight motion which is its 

 dne in that principal oscillation, which can now be sustained indefi- 

 nitely by the other motive work. 



This is the phenomenon remarked by Ellicott in his first expe- 

 riment when the clock n° 2 regularly made n" 1 stop. 



We have now gradually reached case C where c^ and c, are small 

 and where /, and 4 differ but slightly ; this case demands, however, 

 separate treatment, for which reason we shall discuss it later on. 



B. Discussion of the case that l^ and /, differ hut ven/ little, 

 but where c^ and c^ are not small ^). 



Before passing to the case C we shall treat the simpler case now 

 mentioned which Avill lead us to phenomena corresponding to those 

 found by Huygens. 



To this end we put l^ = l^ -\- A, and substitute this in the cubic 

 equation (14j. Then by writing for one of the roots of that equation 

 4 + ff and by treating A and d as small quantities we shall easily 

 find for the length of pendulum of the intermediate principal 

 oscillation the value 



'•+o-fc-^' (IS) 



from which is evident that this length of pendulum divides the 

 distance between /^ and 4 i^^ ratio of c^'' : c^^. 



The two other roots satisfy approximately the quadratic equation: 

 {l'-X)il,-X)-{c,^-\.<-)l'h = .... (19) 



1) This was really observed by Ellicott (1. c, p. 132 and 133) for both clocks, 

 however only temporarily, for at last the work of the first clock came entirely 

 ^0 a stop. Compare for the rest the experiment of Daniel Bernoulli with the two 

 scales mentioned in § 3. 



2) If li is perfectly equal to lo = I, then of com^se (14) has a root x = l for 

 whose principal oscillation according to (15) the frame remains in rest. The remaining 

 roots are found by means of the quadratic equation (^'— A) (^ — },)—{Ci^-{-C2") l'l=0. 

 One of them will nearly correspond to I if q and Co are both small fractions. All 

 this in accordance with Pvouth's solution (I.e. note (1) page 439) which refers 

 exclusively to this case and also to that of Euler (barring what is remarked in 

 that note). 



