( 451 ) 



A 9 



: 



Fig. 5. 



rapid principal osc. 



1 » rapid pendulum l^ 

 ; C interm. principal osc. 

 low pendulum /; 



conditions are represented by Fig. 5, where 

 we have moreover to notice that the third 

 root belonging to the slow principal oscillation 

 differs but little from /'. 



We can now show that for the rapid prin- 

 cipal oscillation as well as for the intermediate 

 one, although not in the same measure, the 

 oscillations of the frame remain small com- 

 pared with those of the pendulums. 



Generally this is already directly evident 

 from the equations (15); this is however 

 not the case when the pendulums are suspend- 

 ed to points of the frame whose horizontal 

 motion is an exceptionally slight one ^), In 

 that case we refer to the general theorem to 

 be proved in the following paragraph, and 

 from which what was assumed ensues im- 

 mediately. 



Let us note before continuing that now 

 for the rapid as well as for the intermediate 



principal oscillation the two pendulums possess amplitudes which 



are mutually of the same order of magnitude. 



i \ reduced system 

 X slow principal osc. 



14. The indicated theorem can be formulated as follows: ivhen 

 the length of pendulum of a principal oscillation approaches closely to 

 h ^f h i^^^'i^^ ^^^0 '^^^ '^^'^"'^ ^/ i^'''^ reduced system, thus a fortiori of 

 the frame alone, is continually small luith respect to that of the pen- 

 dulum correspondhig to l^ or /,. 



To prove this we compare in formula (8) the three terms : 



dx^ • • 

 4 M' u'^ ; m^ k^ —J u' <f^ and h m^ a^^ <f\\ For the proportion of the 

 au 



dx^ ■ , . 

 second to the third can be written '^ ^rj u' : i^g)^, or on account 



du 



of equation (10), 2 -^ u''"'' : /, x, = 2 §/'"^ -.1^x^ = 2 (A — /J : /,. The 



second is therefore, when X approaches l^ closely, small with respect 

 to the third, which can thus be regarded in such a case to represent 

 at first approximation the vis viva of the first pendulum. 



1) That is to say, when the third cause mentioned in note (2) p. 44G has given 

 rise to the sraallness of q and c^. 



^ 



