762 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
points of contact within these planes. The two right lines OH, OH' drawn through the 
point of incidence O and the two points of contact H, H' will be the refracted rays. 
By means of the theorem referred to in the last number I have replaced this con- 
struction by the following one, much easier to manage. Construct with regard to the 
third auxiliary ellipsoid E the polar line of the trace HR. This polar line, which may 
be denoted by SS, meets the wave-surface within the crystal in the two points H and H', 
OH and OH' being, as before, the two refracted rays. 
The plane HOH', containing both refracted rays OH, OH', may be called the plane of 
refraction. There are, generally speaking, four tangent planes passing through RR, as 
there are four points where the wave-surface is intersected by SS. We get therefore 
four rays, all confined within the plane of refraction, but two of them, not entering the 
crystal, are foreign to the question. 
6. The plane of refraction may be constructed solely by means of the third ellipsoid 
E. The details of this construction depend upon the well-known different modes of 
determining the polar line SS. On proceeding in this way we meet some remarkable 
corollaries concerning double refraction *. 
7. The poles of all planes passing through the trace RR, represented by 
qy-fpx=w . . 
( 5 ), 
are points of SS. All right lines passing through the point of incidence O and these 
poles fall within the plane of refraction confining SS. These right lines may likewise 
be regarded as diameters of the ellipsoid E conjugate to diametral planes passing 
through the trace along which the surface of the crystal, i. e. the plane xy, is inter- 
sected by the wave-front in its primitive position, the trace being parallel to RR and 
represented by 
Hence qy+px= 0 (11) 
The plane of refraction is that diametral plane of the ellipsoid E, the conjugate dia- 
meter of which is perpendicular to the plane of incidence in O. 
* In concluding a former paper, “Discussion de la forme generale des ondes lumineuses” (Crelle’s Journal, 
No. xix. pp. 1 & 91, Mai 1838), I gave the following construction: — 
“ Construisez, par rapport a l’ellipsoide directeur, la ligne droite polaire (SS) de celle qui est perpendiculaire 
au plan d’incidence en O'. Elle coupera la surface de l’onde, decrite autour du point 0, en deux points. Les 
deux lignes droites qui vont du point 0 aboutir a ces points seront les deux rayons refractes ; tandis que les 
deux plans, qui, contenant la perpendiculaire en 0' (RR), passent par ces deux m ernes points seront les fronts 
des deux ondes planes correspondantes. Enfin il a ete demontre, dans ce qui precede, que les deux plans de 
vibration sont ceux qu’on obtient en conduisant par les rayons lumineux (refractes) des plans perpendiculaires 
aux fronts des ondes correspondantes.” 
At the present occasion I resume the discussion, announced by myself twenty-six years ago, of a part of this 
construction. More recently, in the eighteenth Legon of his valuable work, ‘ Theorie mathematique de l’Elasti- 
cite’ (1852), M. Lam£ reproduces the curious relation between the wave-surface and the third ellipsoid. He 
presents in the following Legon a remarkable theorem, “ which is one of those immediately derived from this 
relation.” [8] 
