IN PERFECTLY TRANSPARENT MEDIA. 199 



By such reductions it appears that IS,, is a linear function of the 

 nine products of L, M, N with 



Now if we set 



e = 

 we have by (4) and (2) 



a 1 /3 2 - / 8 1 a 2 . (8) 



Therefore IS,, is a linear function of the nine products of L, M, N 

 with LG, M9, NO. That is, IS,, is the product of and a quadratic 

 function of L, M and N. We may therefore write 



= f e = j [L(/3 iy2 - y 1( 8 2 ) + M( yi a 2 - a,y 2 ) + N (<.,& - &,)], (9) 



If (/ 



where $ is a quadratic function of L, M and N, dependent, however, 

 on the nature of the medium and the period of oscillation. 



9. It will be useful to consider more closely the geometrical 

 significance of the quantity 0. For this purpose it will be convenient 

 to have a definite understanding with respect to the relative position 

 of the coordinate axes. 



We shall suppose that the axes of X, Y, and Z are related in the 

 same way as lines drawn to the right, forward and upward, so that 

 a rotation from X to Y appears clockwise to one looking in the 

 direction of Z. 



Now if from any same point, as the origin of coordinates, we lay 

 off lines representing in direction and magnitude the displacements 

 in all the different wave-planes, we obtain an ellipse, which we may 

 call the displacement-ellipse* Of this, one radius vector (pj) will 

 have the components a 1? /3 lt y 15 and another (p 2 ) the components 

 a 2 , /3 2 , y 2 - These will belong to conjugate diameters, each being 

 parallel to the tangent at the extremity of the other. The area of 

 the ellipse will therefore be equal to the parallelogram of which 

 /D! and p z are two sides, multiplied by TT. Now it is evident that 

 &y2~~yA, yi a 2~ a iy2> a A~ a A are numerically equal to the pro- 

 jections of this parallelogram on the planes of the coordinate axes, 

 and are each positive or negative according as a revolution from 

 Pi t Pz appears clockwise or counter-clockwise to one looking in 

 the direction of the proper coordinate axis. Hence, will be 

 numerically equal to the parallelogram, that is, to the area of the 

 displacement-ellipse divided by TT, and will be positive or negative 



* This ellipse, which represents the simultaneous displacements in different parts of 

 the field, will also represent the successive displacements at any same point in the 

 corresponding system of progressive waves. 



