( 444 ) 



centrated and on the other side the entire vis viva of the reduced 

 system multiplied by the distance between suspension point and 

 centre of qravity of the first pendtdum and divided by its length of 

 pendtdum ; and in the same loay c^. 



Discussion of the general case. 



7. Passing to tlie discussion of equation (14) we notice that 

 in the supposition /j > 4 we have: i^(+ go) neg.; F{1,) pos.; F {l^) 

 neg.; 7^ (0) = /'/^ 4 (1 — c^"" — c^'), and tlierefore with reference to 

 (17) where k^ : h and /-^ : ^ < 1, ^'^(0) always positive. 



So there are three principal oscillations. The slowest, which we 

 shall call the sloiv principal one has a synchrone length of pendulum 

 greater than the greatest length of pendulum of both suspended 

 pendulums ; of the intermediate principal one the length of pendulum lies 

 between that of these two pendulums; of the rapid principal one it 

 is shorter than the shorter of the two ^). Further we can note that 

 when /' ]> /i > 4 ^lie length of pendulum of the slow principal 

 one is greater (han /' and that for /^ > /, > /' the rapid principal 

 one has a smaller length of pendulum than /'. 



The following graphic representation gives these results ^) for the 

 case I' y>li'^h^ practically the most important. 



1) This is the case for V positive and this proves that when the reduced system 

 is stable, this must also be the case for the original system with the two suspended 

 pendulums. If /' is infinite, thus tlie reduced system at first approximation in 

 indifferent equilibrium, then the slow principal oscillation has vanished or rather 

 has passed into an at first approximation uniform motion of the entire system, which 

 would soon be extinguished by the friction. The two other principal ones remain 

 and their lengths of pendulum are found out of the quadratic equation: 



For /' negative F (0) becomes negative too, but F (— c^) positive, so then always 

 one of the principal lengths of pendulum is negative. From this ensues that when 

 the reduced system is unstable, this is also the case for the original one. 



2) Of course these results are in perfect harmony with and partly reducible from the 

 well-known theorem according to which when removing one or more degrees of 

 freedom by the introduction of new connections the new periods must lie between the 

 former ones. To show this we can 1. fix the frame, 2. bring about two connections 

 in such a way that the pendulums are compelled to make a translation in a vertical 

 direction when the frame is moved. In the latter case it is easy to see that the 

 time of oscillation of the reduced system must appear. 



For the rest these same results are found back in the main, extended in a way easy 

 to understand for more than two suspended pendulums, in the work of W.Dumas, 

 quoted in note G, page 439 which I did not get until I had finished my investi- 

 gations. By him also the length of pendulum of the reduced system is introduced. 

 However, he has not taken so general as we have done the mechanism of one 

 degree of freedom, on which the pendulums were suspended. 



