708 Proceedings of Royal Society of Edinburgh. [sess. 
of them, and the corresponding points in all the others form a 
homogeneous assemblage of points. If this condition is fulfilled 
for any one chosen point of one body, (a) and ( b ) imply it for any 
other ; and vice versa if this condition is fulfilled for three points 
of one body chosen arbitrarily hut not in one line, ( b ) is a necessary 
consequence. 
(d) A homogeneous assemblage of points means, and cannot mean 
other than, an assemblage which presents the same aspect and the 
same absolute orientation when viewed from different points of the 
assemblage. Some confusion of ideas has been introduced by 
leaving the generalised simplicity of Bravais, and considering an 
assemblage of double points, or triple points, or quadruple points, 
without noticing its being resolvable into two, or three, or four 
similar homogeneous assemblages of single points. 
(e) Rows of Points in a Homogeneous Assemblage. — Through any 
two points of the assemblage draw a straight line, and produce it 
indefinitely in both directions. All points on this line at intervals 
successively equal to the distance between the two chosen points, 
are points of the assemblage. The interval between each point and 
the next to it on either side in the line is called by Bravais the 
parameter of the row. 
(/) Planes of Points (“reseaux”) in a Homogeneous Assemblage. 
— Take at random any three points of the group. The case of 
there being other points of the assemblage on the sides or within 
the area of the triangle of the chosen points may he excluded. 
Along the line of each side of the triangle produced in both 
directions, mark off in succession lengths equal to the side, and 
through each division draw parallels to the other two sides. The 
plane of the triangle extended indefinitely in all directions is 
thus divided into equal and homochirally similar triangles 
turned alternately in opposite directions. At every angle of 
see two tetrahedrons, OPQR, OP'Q'R', which are equal, and allochirally 
similar, being parallel perverts, either of the other, or parallel mutual perverts. 
From every point P of a body or group of points, draw a line through any 
one point 0, and produce to P', making OP' = PO. The group of points (P r ) 
is a parallel pervert of the group (P). The groups (P) and (P') are parallel 
mutual perverts. Turn (P') 180° round any line OK. In the position thus 
reached, it is the image of (P) in a plane mirror through 0, perpendicular to 
OK. In their present positions they are mutual perverts inverted relatively 
to the line OK. Mutual perverts are allochirally similar. 
