426 Dr. Silvanus Thompson on 



Case (ii.). If <£ 5 =0, then 



srn</> 5 =0, cos <j> d =l, 



aud the equation reduces to 



5r _ 20^ 3 16V _ X _ n 

 X X 3 + X 5 ~ Y 5 ~ ' 



corresponding to fig. 12, subject to limitations as before. 



Case (iii.). If 4>*> = Tr, then 



sin $5 = 0, cos $ 5 =— 1, 



and the equation is 



5_£ _ 20^ 16./> ;/ _ 

 X X 3 + X & + Y 5 ~ ' 



which is the equation to fig. 13. 



§ 5. The Higher Terms. Generalized expression for the 

 Lissajous' curve of the nth term. 



The expression for any higher term has the general form 

 G »+(fT- 2G »y .cosfc=sin**„, 



* n- n 



which reduces in the case of symmetry to 



^" + \yJ - 1 ' 



aud 7 in cases where <f> n =0, to 



G„±f =0, 



n 



in which expressions G n is written for the series 



X # 3 x 5 x" — — n? 1 



where 



c= 



h 



( n ^l2)( 7l 2_ 3 2) ( w 2- r _-93) 



where r is an odd number greater than unity. 



§ 6. It is now possible to restate the proposition at the 

 end of § 2 in the following way. Any hysteresis loop can be 



