770 REPORT—1863. 
for the limits of the angle AOB are evidently 60° and 120°; whence the 
perpendicular distance of OA from the parallel nearest to it but one is 
> OBV 3; 1 e, the distance of any node on that parallel from O is > OB. 
~ If then we join any node O, first to a node A, which lies as near to O as 
any other node, and, secondly, to a node B, which lies as near to O as any 
node out of the line OA, the joining lines are adjacent sides of a reduced 
parallelogram ; for, by what precedes, B must lie on one or other of the 
parallels nearest to OA. 
In general, a system of nodes has but one reduced parallelism, because in 
general there is a pair of opposite nodes AA’ each of which is nearer to O than 
any other node whatever, and a second pair of opposite nodes BB’, not lying 
in the line AOA’, each of which is nearer to O than any node not lying in 
that line. Even if A and B are equidistant from O, provided only that their 
common distance from O is less than the distance of any other node from O, the 
system has but one reduced parallelism. But there are two special cases in 
which a nodal system admits of more than one reduced parallelism. 
1. If there is one pair of opposite nodes AA’ nearer to O than any other 
node, and two pairs BB’, 6b’, equidistant from O, not lying in the line AOA’, 
and nearer to O than any other node not in that line, the system admits of 
two reduced parallelisms, having one set of parallels in common, and haying 
their common set of parallels equally inclined to the other two sets. 
2. If there are three pairs of points at the minimum distance from O, 
the system of nodes forms a system of equilateral triangles; and, suppressing 
in turn each one of the three systems of parallel lines by which these 
triangles are formed, we obtain the three reduced parallelisms of which the 
system admits. 
That, in these two cases, the reduced parallelisms are such as we have 
described, and that, except in these two cases, there is but one reduced — 
parallelism, may be inferred from the existence of a reduced parallelogram 
in every system, and from the properties which have been shown to belong 
to it. 
To apply these results to the theory of quadratic forms, let aa*+2bay+cy” 
be a form of the negative determinant —A; let cos as? and with a pair 
of axes inclined to one another at an angle w, let us construct all the points 
whose coordinates are integral multiples of Wa and “¢ respectively ; thus 
forming a nodal system. The expression awv* + 2bay-+cy’ will then represent 
the square of the distance between any two nodes, the differences of whose coor- 
dinates are «Va and yb: and the area of an elementary parallelogram will 
be VA. If the transformation e=aX+/Y, y=yX+6Y,where ad—Py= +1, 
change ax?+2bay+cy” into AX’+2BXY+CY’; and if, in the same plane 
as before, we construct a nodal system corresponding to the latter form—the 
directions of rotation from the axis of X to the axis of Y, and from the axis 
of x to that of y, being the same—it will be found that the two systems may 
be made to coincide. For if we consider the point in the first system whose 
coordinates are w/a, y¢ as corresponding to the point in the second system 
whose coordinates are X./ A, Y/C, the distance between any two points of. 
the first system is equal to that between the corresponding points of the second 
system ; therefore the two systems are identical, and are either similarly situ- 
ated, i. ¢. are capable of being made to coincide by moving either of them 
about in their common plane, or else are symmetrically situated, 2. ¢. are — 
capable of being made to coincide after the plane of one of them has been 
