( 440 ) 



So it seemed worth while looking at the qiiestioji from another side, and 

 studying the behaviour of a very generally chosen mechanism ^) with 

 one degree of freedom, and with two compound pendulums attached 

 to it; noting particularly the case that both pendulums have about 

 equal periods of oscillation, v»^hilst at the same time for the applica- 

 tion of the phenomena of sympathy of clocks the intluence of the 

 motive works will have to be paid attention to. 



Moreover it is worth noticing that the results obtained in this way 

 will also be applicable to the case that the connection between the 

 two pendulums is brought about by means of an elastic mechanism, 

 every time when })ractically speaking only one of the infinite number 

 of mannei'S of motion is operating which such a mechanism can 

 have. Such a manner of motion will have a definite time of oscilla- 

 tion for itself, which will play the same part in the results as if it 

 belonged to a non-elastic mechanism with one degree of freedom. 



Deduction of the equations of motion. 



4. Let S represent for any point of the mechanism with one 

 degree of freedom, to be named in future the "frame", the linear 

 displacement out of the position of equilibrium common to frame and 

 pendulums; let b'"'^ be its maximum value for a definite oscillation to 

 be regarded as equal on both sides for small oscillations ; let ^j and 

 C, be its values for the suspension points 0^ and O.-^ of the pendulums; 

 let M be the mass of the frame; let m.^ and m^ be that of the 

 pendulums; a^ an a^ the radii of gyration of the pendulums about 

 their suspension points; tp^ and (f^ their angles of deviation from 

 the vertical position of equilibrium; .i\, y^ and i\, y, the horizontal and 

 the vertical coordinates of 0-^ and of 0^, h the vertical coordinate of 

 the centre of gravity of the frame; taking all these vertical coordinates 

 opposite to the direction of gravitation. 



So we begin by introducing for the frame a suitable general coor- 

 dinate u, for which we choose the quantity determined by the relation 



Mu'' = ( S' dm • (1) 



where the integration extends to all the moving parts of the frame; 

 this quantity might therefore be called the mean displacement of 

 the particles of the frame. 



1) We assume with respect to Ibis mechanism no other restriction than that the 

 motions of each of its material parts just as those of the two pendulums take 

 place in mutually parallel vertical planes, i.o.w. we restrict ourselves to a problem 

 in two dimensions. 



