THEORY OF VIBRATIONS 49 



oscillates in the several normal modes will in general be different. 

 The superposition then takes place of course according to the 

 law of geometrical or vector addition. 



It will suffice to consider the case of two degrees of freedom, 

 where we have independent simple-harmonic vibrations in the 

 directions corresponding to the Bs l} &s 2 of 15. The result is a 

 plane orbit, usually of a complicated character. For instance, 

 in the case of Blackburn's pendulum ( 14), we have 



x = A cos (nj + ej), y = E cos (n z t + e 2 ), (1) 



where x, y are rectangular coordinates. The orbit is here 

 contained within the rectangle bounded -by the lines x A, 

 y = E. If n lt HZ are commensurable, the values of x, y and 

 x, y will recur after the lapse of an interval equal to the least 

 common multiple of the two periods, and the path will be 

 re-entrant. The resulting figures, obtained in this and in other 

 ways, are associated with the name of Lissajous*, who has had 

 many followers in a region which is very attractive from the 

 experimental point of view. 



The simplest case is that of rij = n^ If we eliminate t in 

 (1) we then obtain 



s(e 1 -e 2 ) + |^ = sm 2 (e 1 - 2 ) (2) 



This represents an ellipse which, if the initial phases e lt e 2 coincide, 

 or differ by TT, degenerates into a straight line (Fig. 20). The 

 simplest mechanical illustration is furnished by the spherical 

 pendulum. When the relation is that of the octave (r^ 2n?) 

 we have a curve with two loops, which may degenerate into one 

 or other of two parabolic arcs (Fig. 21). The curves in these and 

 in other cases of commensurability are easily traced from the 

 formulae (1) with the help of tables. A simple geometrical 

 construction is indicated in Fig. 22, where the circumferences of 

 the auxiliary circles are divided into segments corresponding to 

 equal intervals of time in the two simple-harmonic motions 

 which are to be compounded. If we start at a corner of any 



* J. A. Lissajous (182280). Professor of physics at the Lyce"e St Louis 

 1850 74; rector of the Academy of Chambe'ry 1874 5, and of Besancon 

 1875 9. His chief memoir, Etude aptique des mouvements vibratoires, was 

 published in 1873. 



L. 4 



