116 



From this it is clear that the proper character of the pattern 



is intimately connected with that of the net-plane PP'P"P"> ', 



(or QQ'Q"Q"' t . . . . etc.), which is, in a sense, its very foundation. 



3. It is thus of high importance for the study of homogeneous 

 distribution in general, to examine first the properties of such 

 homogeneous and regular systems of mathematical points. If for 

 the moment we make abstraction from the particular case of the 

 distribution of points in a "plane" only, and if we extend our 

 reasonings to tridimensional systems of points, asking what their 



Fi g? 104. 



arrangement in space must be in order to fulfil the condition of 

 homogeneity as defined above, the answer will evidently be, that 

 these points must be situated at the corners of an infinite number 

 of congruent and contiguous parallelepiped cells, in which space 

 can be divided without leaving any room between them. In this 

 way these points appear situated like the knots of a net- work in space, 

 the meshes of which are the congruent and contiguously arranged 

 parallelepiped cells just mentioned. 



A general type of such network, which bears the name of a space- 

 lattice, - - is reproduced in fig. 104.. Starting with a point 0, we can 

 look for the point P l nearest to it at a distance d lt then for the 

 second nearest point P 2 at a distance d 2 , and finally for the third 

 nearest P 3 at a distance d s from 0. The directions of OP lt OP 2 , and 



