304 ON THE PROPAGATION OF LIGHT 



the equations of an infinite number of ellipses which, in general, 

 do not belong to the same curve surface. If, however, we 

 cause each ellipsis to turn 90 in its own plane, the whole system 

 will belong to an ellipsoid, as may be thus shewn: Let (xyz) be 

 the co-ordinates of any point p in its original position, and 

 (xyz) the co-ordinates of the point p' which would coincide 

 withj? when the ellipsis is turned 90 in its own plane. Then 



since the distance from the origin is unaltered, 



Q = ax' + by' + cz, since the plane is the same, 

 = xx + yy' + zz, since p Op = 90. 



The two last equations give 



x y' s! 



5- = - = f - = w suppose. 

 cy oz az ex ox ay 



Hence the last of the equations (9) becomes 



But 



a-* + 2/ 2 + / = o> 2 {(cy - Izf + (az - ex? + (Ix - ay)*} 



= o> 2 [(b* + a 2 ) z 2 + (c 2 + a 8 ) y z + (b* 4- c 2 ) x* - 2 (bcyz + abxy + acxz] 



Therefore e a = l, 



and our equation finally becomes 



l = Lx'* + My" + NJ* .................. (10). 



We thus see that if we conceive a section made in the 

 ellipsoid to which the equation (10) belongs, by a plane passing 

 through its centre and parallel to the wave's front, this section, 

 when turned 90 degrees in its own plane, will coincide with a 

 similar section of the ellipsoid to which the equation (8) belongs, 

 and which gives the directions of the disturbance that will cause 



