226 



UNDULATORY THEORY OF LIGHT. 



will generally not be in the plane of incidence ; nor will the ray or radius of the 

 wave surface be normal to the tangent plane. 



The three principal sections of the wave surface present each two curves 

 returning into themselves, as shown in these figures : — 



c 



Fig. 66. 



The equation of the section through a b, Fig. 66, is deduced from the general 

 equation of the wave surface, by putting z=z{), when it becomes — 

 (a V + hhf) {x^ +if-{a?+lr) + a'a? + bhf + a^b\^=Q, 

 which may be resolved into the two foctors — 



{a^x^ + bhf-a?b''){^+y'-c'')=Q, "[51.] 



being the equation of an ellipse and a circle combined. In like manner, making 

 X nothing, we obtain the equation of the section through b c, Fig. 65 — 



(j2^2_^^2;j2_S2c2)(2/2 + ;22-a2)=0 J [52.] 



and making y nothing, that of the section through a c, Fig. 67 — 



{a^3?+c^z^-a\^){x^-{-z^-b'')=0. [53.] 



This last section is remarkable, as showing an intersection of the circle and 

 ellipse. The intersection is necessary, because the diameter of the circle is the 

 mean axis of elasticity =:J, while the major and minor axes of the eclipse are the 

 extreme axes of elasticity, a and c. The points of intersection, shown at N, W, 

 &CC., are the inosculating points of the two nappes of the wave surface. 



Since the velocity of ray propagation is measured by the radius of the wave 

 surface, it is evident that, along the radii drawn to N, N', &c., there may be two 

 refracted rays having the same velocity. These lines have a peculiar optical 

 interest. Their inclination to x, or a, the axis of greatest elasticity, (or the 

 angle MCN) may be found from the equation (putting i3z=MCN,) 



^ MN MN 



MN and NO are obtained by making both fjictors of the equation of the 

 section, just given, simultaneously=0. The values of x and z which render 

 this possible are the values of NO and NM. We have then, 



x'+z''—b"=0, and a'x'^+c'^z^—a'c'^O. 

 from which we obtain, by elimination. 



aP-- 





Whence, 



MN z , aV b- — c2 



^c77^=-^==c — 7 ^r=tana, 



NO X cVa'—b' 



[54.] 



which differs a little from the value found for the tangent of inclination of the 

 normals to the circular sections. But these normals are the directions of equal 

 wave velocity; and CN is the direction of equal ray velocity. These two direc- 

 tions are therefore not coincident, though nearly so. 



