Hysteresis Loops and Li&sajous 1 Figures. 

 Fifth Term (Fifth Harmonic) ; n=5. 

 The two equations now are 



- r = sin #, 



But 



y_ = sin(50 + <£ 5 ), 



= sin 50 . cos 05 + cos 50 . sin 5 . 



sin 5(9 = 5 sin (9-20 sin 3 + 16 sin 5 0. 



425 



(U 



(2) 

 (2a) 



Inserting for sin its value in (1), we get 



sin 5$ = 



X 



20a, 8 15«r 

 "X 3 " + ~X 5 ~ 



Substituting this value in (2 a), we deduce 

 ^ x - - -^rg- + -^5- ) COS 5 — ^r = — cos 50 . sm 0= 

 Also multiplying (3) by sin 5 , we have 



\X — ST "X 5 "/ sm < r 5 = sin 5 ^ • sin t* 5 ' 



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

 20^ 3 16.v 5 \ 2 ^f 2j\ 2 _ 



5/ \X 



(3) 



(4) 



(5) 



/ox 20.c 3 16.i' 8 V /j/V „/5a; 20a; 3 16« 5 \ « 



This is the equation of the general Lissajous' figure of the 

 fifth order, representing the result of compounding two 

 vibrations having relative frequencies of 1 : 5, and a general 

 form like fig. 10 (PL VI.). 



As before, three cases arise : — 



Case (i.). If <f> 5 = ^7T or J-jt, then 



sin 0- = ± 1 an d cos 5 = 0, 

 and then the equation becomes 



which is symmetrical as in fi"\ 11. 



