464. J. W. Gibbs— Double Refraction and Circular 
of s,, that we have to consider. S, is therefore a linear 
ee of the space-averages of these nine quantities. But 
by (3 
eo at, (12-282), 
da? dae 
and the space-average of this, at a moment of maximum 
displacement, is by (1) 
aL 
(Buy sB). 
By such reductions it appears that /S,, is a linear function of 
the nine products of L, M, N with . 
BiV2—ViPr  ViG2—NHYr, fi— Byars. 
Now if we set 
O=L (fiy2—y1h2)+M (yia,—ay2) +N (af.—fia,), (7) 
we have by (4) and (2) 
LO=fh,\y.—yV:f, MO=y,a,—My, NO=a,f,—f,a, (8) 
Therefore /S, 6 i linear cade of the nine products of L, M, 
with L6@, M 6. eae s, /S,, is the product of 6 and a 
quadratic function of L,Ma dN.’ We may therefore write 
8,=5 O=F{L(A,y.— yuh.) +M(y,a.— ay.) +N (a, f.— 3,4) |, (9) 
where @ is a ce uadratic epee of L, M and N, depende nt, 
the nature of the medium and the period of 
It will be useful to consider md closely t the geometrical 
significance of ae quantity 6. For this purpose it will be con- 
ent to have a definite Se ceuinatis with respect to the 
rolativd poston of the coordinate axes. 
_ We sh Ages be the axes of X, Y, and Z are related 
du 
lay ofl lines representing - in direction and nage the dis- 
placements in all the a at wave-planes, we obtain an 
ellipse, which we may call t displacement-ellipse.* OF 1 this, 
one radius vector (,) will Ses the components 4, f,, 7;, and 
* This ellipse, which represents the simultaneous Dopamine cr in different 
of the field, will also represent the successive displacements at any same 
point in the corresponding system o 
