( 44J ) 



■ Foi' small oscillations of the frame we can put : w=?2w('"), ^=?z§('")^ 

 where n is a function of time, but the same for all the points of 

 the fj-ame. 



So we have for such vibrations : 



Mu' 



: I (w5(»0)^ dm = f 



Af{nu^^'^)y = (w?(»0)^ dm z= \ ^^ dm ; 



so that ^Mu^ proves to represent the vis viva of the frame. 



For the vis viva of the first pendulum we find, if k^ denotes the 

 distance between its suspension point Oj and its centre of gravity, 

 and if (f^ is reckoned (like (f^) in such a way that a positive value 

 of y>i increases the horizontal coordinate of the centre of gravity : 



k [^1 ^i" + 2 Wj /Cj .f 1 '( 1 + mi 01=* (/ij'] = 



therefore for the entire vis viva of the whole system : 



u^ + i Wj a^' yi'' -f ^ m, ttj" y,' + 



m. 



d;,\. . dx^. 



du J du 



^ 2 



^I + ^«1 ( -^ ) + w^ 



du 



dx^ • . i/.-Tj . . 



-^m^h^—-u <f^ + m, ^', -7- ?^ ^3 ; 

 aw aw 



(2) 



and further for the potential energy ^) 



V=^\g 



' d'h d'y, 



du^ du^ 



du^ 



u^ + h ^^h oK^fi" -\-km^9 K ^^' • (3) 



M 



5. To simplify further we introduce the new variables' determined by: 



d^ 

 du 



y du 



m„ 



= ^.^+7/1,?," + ^,?,^ (4) 



where 



if' = if + m^ + m, , (5) 



represents the entire mass of the whole system ; this variable u' 



is proportional to 21, because for small vibrations — ^ and — -, as 



du du 



indeed all such derivatives appearing in the formulae, may be regarded 



as constant. 



Ï) Indeed that potential energy amounts to Mgh-\-migi/i -\- m^gyz — Wigky cos (pi — 



— mz gkci cos rg + a constant. By developing according to 11, taking note that on 



... ,^ dh dy^ dy 



account of the equihbrium M - — \- mi 1- m^ - — is equal to and by proper 



du du du ^ r J 



choice of constant, we can easily deduce (3) from it, 



