( 442 ) 

 Out of this proportionality follows easily 



M' u" = 



^'^+^"^'^)+"^(è 



7> = Mu' -{ mj^' ^ mj^' . (6) 



which proves that | Af'u'^ represents the vis viva of what we shall 



call the rexliiced system, which system consists of the frame and of the 



masses of the pendulums each transferred to the corresponding 



suspension point 0^ or 0^. 



If now likewise we introduce the vertical coordinate A' of the centre 



of gravity of the reduced system, so that M'h' = Mh -\- m^ y^ -\- m^ y^, 



d-h' 

 the first term of (3) transforms itself into \ g M' — — u'^, for which, 



however, on account of the mutual proportionality of u and u' we 



cPh' , 

 may write : \ g M' -— w '. 



So for the reduced system it holds that T =z ^ M' u"^ and F' = 



crh' , , . , . ,. . 



z=z \g M' — y- ir ; if now^ we write for this system the equations ot motion, 



and if we then introduce the length /' of the simple pendulum which 

 is synchrone to this system ^) we shall easily find : 



f^ = (0-' (V) 



ClU 



Thus we tinally may write for (2) and (3) : 

 T = ^,3I'u" 4- \^n^<<k" + h^n^<-<f." + ^^hh -^ ^"''t, + m,h, '^ u' if,; . (8) 

 V=yM'{I!)-U^-^im,gL<f^ + im,gk,<f^ ... (9) 



Application of the equations of Lagrange and substitution of the 

 expressions : 



u' = rt'^'"^ sin \/^t; <f, = v., sin [/jt; r/>, = y., sin [/ y t . . (10) 

 /. /• ^ 



leads further easily to the equations 



M' H' - I) u'^"'-' 4- ^n,k,l' ^ X, + m,kj ^ ;c, = ; . . (11) 



du du 



1) Should the reduced system be in indifferent equilibrium as was probably the 

 case in Ellicott's experiments /' is infinite ; if it were in unstable equilibrium this 

 would correspond to a negative value of /'. We shall again refer to these cases 

 in the notes. In the text we shall always consider I' positive, hence the reduced 

 system stable. 



