( 446 ) 



been taken to decrease siifTiciently the frictions in the frame. However 

 in sucli a case the beliaviour of the two clocks would differ greatly from 

 what was observed concerning the phenomena of sympathy; and in 

 the more probable supposition that the motive works will prove to 

 be unable to sustain a considerable motion of the frame, which 

 motion would absorb a great part of the energy, each of the principal 

 oscillations as well as each combination of them will after a certain 

 time have to come to a stop. 



So we shall leave this general case, and pass to the discussion of 

 three special cases, which are more important for the consideration 

 of the phenomena of sympathy, namely A the case that /^ and ^ 

 differ rather much, but where c^ and c^ are small numbers, B the 

 case, that /^ and /^ differ but little, but Cj and c^ are not small, 

 C the case where I^ and 4 differ but little and c^ and c^ are 

 both very small. In all these discussions we shall suppose /' ]> /j > 4 

 and /' differing considerably from /^ and I^. Tiie treatment of other 

 special cases, e.g. Cj small but c^ not, will not furnish any more 

 difficulties if such a mechanism were to present itself ^). 



A. Discussion of the case thai l^ and 4 differ rather much hut 

 where c^ and c^ are small ^). 



In this case F {I'), F (l,) and F(l^) are all very small, from 

 which is evident that each of the three roots of equation (14) is 

 closely corresponding to one of these three quantities, so that the 

 graphic representation of Fig. 2 looks as is indicated in Fig. 3. 



From this then ensues according to (15) that for the rapid principal 

 oscillation the oscillations of the rapid pendulum are much wider 

 than of the slow one '), and that for the intermediate principal oscillation 



1) Also the case Z' = oo differs in nolhing, as far as tlie results are concerned, 

 from the cases treated here but by the vanishing of the slow principal oscillation. 



2) The smallncss of each of these coefficients may according to (16) be due to 

 three different causes, namely 1. to the smallness of ki : /j which will not easily 

 appear in clocks, 2. to the fact that the masses of the pendulums are small 

 with respect to that of the frame, 3. to the fact that the pendulums are suspended 

 to points of the frame whose horizontal motion is a slight one compared to that 

 of other points of that frame. It is remarkable that this difference of cause has 

 hardly any influence on the considerations fohowing here, and therefore on the 

 phenomera which will present themselves. 



^) Then still when in (15) fo'''"^ might prove to be very small compared to fi*^"") ; 

 for as a first approximation for I^ — A we find: Cci"l'L:{l' — L), and therefore ^3 = 

 — M'{1' — L) {ii'^'"^)":mo k, I' ^o^'"). So the motion of the frame determined by w'^*"^ is slight 

 compared to that of the rapid pendulum and consequently x^ is small compared to ^2. 



