306 ON THE PROPAGATION OF LIGHT 



Similarly, if the plane of the wave's front is parallel to the 

 plane (xz), the wave- velocities are 



. The disturbance being parallel to the axis x, 



to the axis z. 



Or if the plane of the wave's front is parallel to (xy), the 

 velocities are 



. The disturbance being parallel to x, 



......................................... y. 



Fresnel supposes that the wave-velocity depends on the 

 direction of the disturbance only, and is independent of the 

 position of the wave's front. Instead of assuming this to be 

 generally true, let us merely suppose it holds good for these 

 three principal waves. Then we shall have 



N+A = C+L, M+A=B+L, and B + N 

 or we may write 



A L = B M=CN=v (suppose). 

 Thus our equation (8) becomes, since a 2 4- 5 2 + c 2 = 1, 

 1 =fji (ax + ly + cz)* + v (y? +3/ 2 + ^ 2 ) 

 + (La 3 + Mb* + Nc*) (x* + 2/ 2 + * 2 ) 

 + L(cy- Izf + M(az- ex) 2 + N(lx- ay)\ 



But the two last lines of this formula easily reduce to 



(M+ N)x* + (N+ L) f+(L + M) z* 

 + L {aV - (by + c*) 2 } + M {% 2 - (ax + cz) 2 } 



And hence our last equation becomes 



1 = (p + M + JST) x* + (v+N+L) y*+ (v+L+ M ) ^+/x (ax+ly+cz}* 

 +L {aV - (by + cz}*} +M{b*if- (ax + cz)*} + JV {cV- (ax -f %) 2 } 



........................... (11). 



In consequence of the condition which was satisfied in 

 forming the equation (8), it is evident that two of its semi- axes 



