Hysteresis Loops and Lissajous' Figures. 423 



and the equation becomes 



or 



y= 



which is a straight line sloping the reverse way as in fig. 5. 



For all other values of </> the ellipse takes some intermediate 

 form. The sine-component of the first term in the harmonic 

 analysis of the current curve corresponds to the orthogonal 

 ellipse ; the cosine-component to the oblique line form. 

 All the intermediate forms of the ellipse could be obtained 

 by wrapping a sine-curve of period T around a cylinder of 

 diameter T/tt and projecting in appropriate directions upon 

 planes parallel to the axis of the cylinder the apparent outline 

 of the sine-curve. 



Third Term (Third Harmonic) ; n = 3. 



Here the two equations are 



x 



^ = sin 0, (1) 



|- = sin (30 + &), • • (2) 



= sin 30 . cos </> 3 + cos 30 . sin <f> 3 . . . (2 a) 

 But 



sin 30 — 3 sin 0—4 sin 3 0, 



by known trigonometrical relations. 



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



• on 3fl ±x* 

 sm 30 = r - X 3 (3) 



Substituting this value in (2 a), we deduce 



(3# 4tt ,3 \ y 



X~~ X3-) COS 03— y = -cos30.sin</> 3 . . . (4) 



Also multiplying (3) by sin <£ 3 , we have 



(Sx 4x\ . 



\X ~W/ sm 93=f m 30. sin 03. ... (5) 



