212 M. Lorenz on the Theory of Light, 



the homogeneity in different directions, must alter at the same 

 time. Thus by a vertical pressure the vertical dimension will be 

 diminished, and thus all the small irregularly lying layers of 

 which the body consists will become horizontal ; the body must 

 then behave like a doubly refracting crystal with one vertical 

 optic axis. 



III. Integration of the Differential Equations : Circular Polari- 

 zation. 



In the foregoing calculations we have come upon rectilinear 

 vibrations only, and we should not be able to deduce any others 

 by further approximation. But, as already mentioned, the calcu- 

 lation presupposes that 3, 4, ... &c. of the magnitudes p p can 

 acquire no constant value by addition and subtraction. This 

 possibility, as the more general case, must nevertheless neces- 

 sarily be taken into consideration; and it then becomes apparent 

 that the formulae thus generalized comprehend also elliptic vibra- 

 tions dependent on the uneven powers of the small quantities a p . 



Retaining our previous notation, and likewise putting 



S = sm(kt—lx—my—nz), 

 S(p) = sin (kt — lx — my—nz + p), 

 the expression (2) assumes the following more general form : — 



l=FoC+2|(± ft )C(± ft )+...+?' S + 2f(±ft)S(±p„) + ... ) 

 where the coefficients of S, S( + p p ) are denoted by accents. The 

 components tj and ?may be determined in an analogous manner. 

 It can now be proved that, in place of the equations (B), the 

 following will be obtained : — 



« 2 , i?o + 2 , 2 r] + a 2> 3^* + b 2} ^ -f b 2> 2 rj' + h, a ?'o = 7r(Vo), f (C) 



«3, ifo + 03, 2V + 03, 3 £) + &3, ll'o + #3,2^0 + #3,3?0 = w(&) • - 



These three equations give rise to three others again, which are 

 formed from them by putting |Wo, go in place of | , y , £~ , and 



— Zo> —Vo> -?o in P lace of ?'o> v'o, So- For if equation (12) is 

 differentiated in regard to kt, G is transformed into — S, and S 

 into C, which is the same thing as changing all the fs on the 

 right-hand side of the equation into £'s and all the £'s into — fs. 

 If, however, f, tj, \ satisfy the differential equations (A), their 

 differential coefficients with respect to kt would do so likewise, and 

 hence it will always be allowable to make the specified changes 

 in the derived equations. 



It can be proved further, that while the previous relations (5) 



