Fi a 
ON THE THEORY OF NUMBERS. 771 
turned over and applied again to the plane of the other. On comparing any 
two corresponding triangles in the two systems, for example the triangle 
obtained by giving to X and Y the values (0, 0), (1, 0), (0, 1), with the 
triangle obtained by giving to w and y the values (0, 0), (a, y), (G8, 6), it 
will be seen that the two systems are similarly or symmetrically situated, 
according as ad—By=-+1, or=—1. 
It thus appears that a class of quadratic forms of a negative determinant 
may be considered to represent a nodal system, and that each form of the class 
corresponds to a parallelism of the system. Conyersely, to each parallelism 
of the system a form of the class corresponds. For let Ox, Oy be lines of 
any parallelism of the system, and OX, OY lines of any other parallelism, 
the directions of rotation from Ow to Oy and from OX to OY being 
the same; let also /a, /¢ be the lengths of the sides of an elementary. 
parallelogram in the first system, and va the cosine of the angle between 
: ¢ 
them ; and let / A, VC, =, have the same signification with regard to 
the second system; then, if (w/a, yc), (XW A, YC) are the coordinates 
of the same node P, we must have two equations of the form «=aX +,/Y, 
y=yX+6Y, in which « and y are integral if X and Y are so, and vice versa; 
hence a, 3, y, 5 are integral, and 2iA—By=+1; the sign of the unit being 
determined by the supposition we have made as tu the situation of the axes 
with respect to one another. Also OP?=aa?+2bxry + cy?=AX?+2BXY + 
CY’; or the two given parallelisms are represented by two properly equiva- 
lent forms. 
The theorem that in every nodal system a reduced parallelism exists, has 
for its arithmetical expression, “In every class a form exists in which 
[26] < [a], [2b] <c.” We thus obtain an independent proof of the theory 
of reduction of Art. 92; the geometrical signification of the special condi- 
tions in the definition of a reduced form is as follows :—If a=c>([2b], the 
corrresponding nodal system has only one reduced parallelism ; but either of 
the two directions in this reduced parallelism may be taken for the axis of x, 
consistently with the condition that the rotation from Ow to Oy should have 
a given direction; the condition 24 >0 implies that if the angle between the 
axes is not right, that direction is to be assumed for the axis of « which ren- 
ders the angle between Ow and Oy acute, Similarly, if a <c, but a=[2b], 
the system has two reduced parallelisms, and the condition 2b>0 distin- 
guishes one of them from the other. If a=[2b]=c, the system has three 
reduced parallelisms, which are identical and similarly placed; the condition 
2b>0 does not distinguish between these, but only between the two modes im) 
which any one of them can be taken. 
The number of automorphics of a class may be ascertained by causing the’ 
nodal system which represents it to revolve in its own plane round one of its 
nodes, and examining the number of positions in which it coincides with its ori- 
ginal position. After a revolution of 180° it will always do so; but in order 
that it should do so in any other position, the first and second sides of its reduced 
parallelogram must be equal, and must include an angle of 90° or 60°, 2. @. 
the system must be one of squares or of equilateral triangles. Hence we infer’ 
(Art. 90) that there are in general but two automorphics for a form of a 
negative determinant, but that for the classes containing the forms #*+y? 
and 2a*+ 2vy+2y? (or multiples of those forms) there are four and six respec- 
tively. 
; 3D2 
