132 Mr GREEN, ON THE PROPAGATION OF LIGHT 



( (du dv dw\ 2 

 • + M {{die + Ty + rf^J 



T \(dv dw\ 2 dv dw\ 

 \ \dz dy ) dy d%\ 



__( (du dw\ 2 du dw\ 

 \ \dz dx 1 dx dz J 



,-{ /du dv\ 2 du dv\ 

 \ \dy dx) dx dy)' 



which, when = A, = B, = C, reduces to the last four lines. 



Making the same substitution in the equation (7), we get 



1 = n (ax + by ■+ ex'f, 

 + (A a 2 + Bb 2 + Cc*) {x 2 + f+ *"), (8.) 



+ L(cy - bzf + M(a% - cx) 2 + N{bx - ayf. 



Let us in the first place suppose the system free from all extra- 

 neous pressure. 



Then A = 0, B = 0, C = 0, 



and the above equation combined with that of a plane parallel to the 

 wave's front will give 



= ax + by + ex (9.) 



1 =L(cy - bzf + M(a% - ex) 2 + N(bx - ay) 2 , 



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 j) in its original position, and (x'y'%) the co-ordinates of the point 

 p which would coincide with p when the ellipse is turned 90° in its 



own plane. Then 



x 2 + y 2 + z 2 = x' 2 + y' 2 + as' 3 , 



since the distance from the origin O is unaltered ; 



= ax' + by' + ess', since the plane is the same; 

 = xx' + yy + zz, since pOp'= 90°. 



