THEORY OF VIBRATIONS 



51 



one of the rectangles in the figure, and proceed diagonally, we 

 pass through a succession of points, equidistant in time, on a 

 curve of the system. 



Fig. 22. 



Another conception of these figures, also due to Lissajous, 

 may be mentioned. If we write 6 for nj, and adjust the origin 

 of t, the formulae (1) are equivalent, on the hypothesis of 

 commensurability, to 



y 



-(# a), 



.(3) 



where p/q is a fraction in its lowest terms. These equations, 

 when combined with 



z asinO, ........................ (4) 



represent a curve of sines traced on the surface of the circular 

 cylinder 



x * + 2 * = a * ........ ................ (5) 



and going through its period p times in q successive circuits 

 of the cylinder. The Lissajous curve (3) is the orthogonal 

 projection of this curve on a plane (z = 0) through the axis of 



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