142 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



over pulleys. The lengths of the strings being chosen so that 

 m 3 is at the axis when the rods are horizontal, m^ and m 2 when 

 the rods are vertical, we must have, if AC = y l} EG = y z , 



For the actual construction of the model, the reader is referred 

 to Boltzmann, Vorlesungen uber die MaxweWscTie Theorie der 

 Electricitat und des Lichtes. 



By means of these models all the properties of Cyclic Systems 

 may be illustrated, and all the phenomena of induction of currents 

 imitated, as will be described in Chapter XII. 



72. Hamilton's Principle the most general dynamical 

 principle. We have seen in this chapter how by means of 

 Hamilton's Principle we may deduce the general equations of 

 motion, and from these the principle of Conservation of Energy. 

 As Hamilton's Principle holds whether the system is conservative 

 or not, it is more general than the principle of Conservation of 

 Energy, which it includes. The principle of energy is not 

 sufficient to deduce the equations of motion. If we know the 

 Lagrangian function we can at once form the equations of motion, 

 and without forming them we may find the energy. For we 

 have 



L = T-W, 



E=T+W. 

 Accordingly 



so that the energy is given in terms of L and its partial deri- 

 vatives. If on the other hand the energy is given as a function 

 of the coordinates and velocities, the Lagrangian function must 

 be found by integrating the above partial differential equation, 

 involving an arbitrary function. In fact if F be a homogeneous 

 linear function of the velocities, the above equation will be 

 satisfied not only by L but also by L + F. For, F being 

 homogeneous, 



