424 Dr. Silvarius Thompson on 



Squaring (4) and (5), and adding, we get 



This is the equation to a figure having the general form of 

 fig. 6 (PI. VI.). which is indeed the well-known Lissajous' 

 figure, compounded of two vibrations the frequencies of 

 which are as 1 : 3. It could be obtained by taking three 

 complete sine-waves, each of period T/3, and wrapping them 

 around a cylinder of diameter T/tt. 



Again there arise three cases : — 



Case (i.). If 03=i7r or |7r, then 



sin 03 = + 1 and cos 3 = 0, 

 and then the equation becomes 



(i-¥Wf,r- 



Here the figure is symmetrical with respect to the axes, 

 as in fig. 7. It is, for the third term, what the orthogonal 

 ellipse is for the first term. 



Case (ii.). If 3 = O, then 





sin 03 = 0, 



COS 03=1, 



d the equation reduces to 





?>x 4# 3 

 X X 3 



-X- =0. 



Y 3 



Here the trilobate form has shrunk to the form of the 

 curved line (fig. 8) precisely as the ellipse shrank to the 

 oblique line of h'g. 4. This line is subject to the limitations 

 that x and y cannot exceed X and Y s , respectively. 



Case (iii.j. If 3 = 7r, then 



sin 03 = 0, cos 03 =— 1, 



and the equation becomes 



3^_4^ ij__ 

 X X 3 + Y 3 - U? 



the graph of which is fig, 9. 



