On the Cause of Elliptical Polarization. IS 



and this, when the waves move only in the direction of x po- 

 sitive, becomes ^ gin {nt-kx^b). 



Now the equations(l.) being of the first degree, may be satis- 

 fied not only by the second members of the equations (?.)» but 

 by the sum of any number of functions of the same form. It 

 follows, therefore, from the observations just made, and from 

 the circumstance that the origin of x is arbitrary, that the 

 complete integrals of the equations (1.) may be expressed by 



yi— X {a^s\n (njt—kx) + «^^ sin (n^^t—kx)} , 



^ = :§' {pia^s\n{nit — kx — b)-^p^^aiiSin{nfit—kx—b)}: ^ '^ 



provided that the waves travel in the direction of x positive, 

 and that the displacements ij and ^ are functions of x and t. 



Suppose the arbitrary coefficients in the equations (8.) to 

 be all zero except a^ , then 



>j = a^ sin {n^t-kx), . . 



^ = p^ a, sin (n^t—k x—b) ^ '' 



= p^a^ {sm {n^t— k x) cos b-^smb cos {n^t—kx)} ', 

 therefore 



(J— Pi «; sin (n^t-^kx) cosbf =z p^^ a^'^ sin^bcos^ {n,t—kx) 

 = §/ a^* sin® b ( 1 — sin {n,t—kx)): 

 hence we have 



an equation to an ellipse of which >j and ? are the coordinates. 

 Consequently, when the system is in a state of motion which 

 can be expressed by the equations (9.)> every molecule de- 

 scribes an ellipse round its place of rest ; and the equations 

 (8.) show that the general motion of the system is equivalent 

 to a number of coexisting motions of the same kind. 



In a future paper I purpose to apply these formulae to the 

 case of elliptical polarization produced by quartz crystaL 



I am. Gentlemen, yours, &c. 

 Litdemoor, Clitheroe, Nov.24, 1837. John TovEY. 



P.S. To justify our neglect of the displacements f, it may 

 be well here to observe, in addition to what has been re- 

 marked (vol. ix. p. 421.) that when x is taken, as we have 

 supposed, perpendicular to the wave- surface, neither the dis- 

 placements >3, ^, nor their differences A>), A ?, cause any change 

 of density in the medium; but the differences Af imply a 

 change of density. If then we suppose the force by which 



