1865.] Prof. Pliicker on a New Geometry of Space. 57 



If we proceed to complexes of the second degree, the field of inquiry is 

 immensely increased. Here any given point of infinite space is the vertex 

 of a cone of the second order, and likewise within any given plane there 

 is a curve of the second class enveloped by rays of the complex. The 

 whole of the infinite number of cones, as well as of the infinite number of 

 enveloped conies, is represented by a linear equation, between the five ray- 

 coordinates r, s, p, a and (spra). The general analytical theory of con- 

 tact may immediately be applied to complexes of the second order, touched 

 by linear complexes, &c. 



In order to elucidate the geometrical conceptions explained, I thought 

 it proper to present, in Section II., an application to optics, leading to a 

 complex of a simple description. Rays of light, constituting in air a 

 complex, will likewise do so after being submitted to any reflexions or 

 refractions whatever. Let us, for instance, suppose that the complex in 

 air is of the first order and its constant equal to zero ; i. <?., that its 

 rays start in every direction from all points of a luminous right line. 

 Let these rays enter a biaxal crystal by any plane surface. Let the 

 luminous line and this surface be perpendicular to each other. Then, 

 within the crystal, the double-refracted rays constitute a new complex, 

 which is represented, like the primitive one, and independently of it, by 

 means of an equation between ray-coordinates. 



For this purpose I return to a paper of mine of the year 1838, concern- 

 ing double refraction, at the end of which, after having mentioned the 

 application of Huyghens's principle to Fresnel's wave-surface and the 

 construction of Sir William Hamilton, I proposed a new construction of the 

 double-refracted rays in the most general case. Here I first made use of 

 an auxiliary ellipsoid, with regard to which the polar plane of every point 

 of the wave-surface is one of its tangent planes, and, reciprocally, the pole 

 of every plane touching the surface one of its points. In representing 

 Fresnel's ellipsoid by the equation 



the new auxiliary ellipsoid may be represented by 



and replaced, for most purposes, by the similar one, 



The construction, as far as we are concerned here, may be expressed thus : 

 Construct at the moment when Fresnel's wave- surface is formed the polar 

 line of the trace along which the surface of the crystal is intersected by the 

 elementary wave. The two refracted rays meet the wave-surface in the two 

 points where it is intersected by the polar line constructed. In the paper of 



