ON PHYSICAL LINES OF FORCE. 



Multiplying the last two equations together, we find 



511 



(151) 



an equation quadratic with respect to m s , the solution of which is 



9/W 3 



m' = 



< 



These values of m' being put in the equations (150) will each give a ratio 

 of A and B, 



A _ a* 6 2 T v(a 2 ft 2 ) 2 + 4w 2 c 4 

 ]B = 2nc* 



which being substituted in equations (149), will satisfy the original equations 

 (148). The most general undulation of such a medium is therefore compounded 

 of two elliptic undulations of different eccentricities travelling with different 

 velocities and rotating in opposite directions. The results may be more easily 

 explained in the case in which a = 6 ; then 



and A= ^-B (153). 



Let us suppose that the value of A is unity for both vibrations, then we 

 shall have 



/ nz \ I nz \ ' 



x = cos nt , I + cos I nt . } 



\ Va 2 ncv \ v +nc 2 / 



I nz \ , . / . nz N 



y = sin ( nt -,-=== j + sin I nt -j=f= 



The first terms of x and y represent a circular vibration in the negative 

 direction, and the second term a circular vibration in the positive direction, 

 the positive having the greatest velocity of propagation. Combining the terms, 

 we may write 



x = 2 cos (nt pz) cos qz\ 



y = 2 cos (nt pz) sin qz] 



n n 



.(156). 



n n 



and 



