80 
axes. Since the notation is the same in either case, it follows 
that the expressions for the geometric relations between faces 
and zones, in terms of their respective indices, will be absolutely 
the same whether we refer the faces to three zone-axes, or their 
rays to the three corresponding rays as axes of coordinates. 
6. Instead of deducing the properties of a system of rays 
referred to three rays as axes, from those of faces referred to three 
zone-axes, it will be better to investigate the properties of Grass- 
manu’s system of rays without any reference to the crystal it is 
intended to represent. Having once established the properties 
of a purely geometric system of rays we may proceed to the 
consideration of crystals, avoiding the mistake sometimes made 
of anticipating the result of a geometrical investigation in the 
enunciation of a physical law, and assert that measurements of 
the angles between normals to the faces of crystals show that 
these normals are subject to the law according to which the 
system of rays was constructed, and that, consequently, all the 
geometrical properties of such a system of rays are properties of 
the rays drawn from any point within the crystal normal to its 
faces. 
The analytical investigation is followed, first, by one in which 
ordinary geometry is used ; and, afterwards, by an investigation of 
the properties of a system of points cn the surface of a sphere, 
the points being the intersections of the rays with the surface of 
a sphere described round the origin of the system as a centre. 
Those propositions are omitted the investigation of which is the 
same whether we employ the polyhedral solid or its reciprocal 
figure. 
ANALYTICAL INVESTIGATION OF THE PROPERTIES OF A 
System oF Rays. 
Rays. 
7. Let OA, OB, OO be any three straight lines not all in 
one plane; a, 6, y any three straight lines in a given proportion ; 
