§1] 
ILLUSTRATIONS FROM ANALYTICS 
3 
but a more general fact will be obtained in § 4 without tedious 
multiplications. Thus a+c and d = ac—b 2 are invariants of 
/, and also of 5, under every transformation of type T. When 
5 = 0 represents a real conic, not a pair of straight lines, the 
conic is an ellipse if d>0, an hyperbola if ¿<0, and a parabola 
if d = 0. When homogeneous coordinates are used, the classi 
fications of conics is wholly different (§ 13). 
If x and y are the coordinates of a point referred to rectan 
gular axes and if x' and y' are the coordinates of the same 
point referred to new axes through the new origin {r, s) and 
parallel to the former axes, respectively, then 
t: x = x'-\-r, y=y’-\-s. 
All of our former expressions which were invariant under 
the transformations T are also invariant under the new trans 
formations t, since each letter a, b, . , . involved is invariant 
under t. But not all of our expressions are invariant under 
a larger set of transformations to be defined later. 
We shall now give an entirely different interpretation to 
the transformations T and t. Instead of considering (x, y) 
and (x', V) to be the same point referred to different pairs 
of coordinate axes, we now regard them as different points 
referred to the same axes. In the case of t, this is accomplished 
by translating the new axes, and each point referred to them, 
in the direction from (r, s) to (0, 0) until those axes coincide 
with the initial axes. Thus any point (x, y) is translated to 
a new point (x', y')> where 
x'=x—r, y' = y—s, 
both points being now referred to the same axes. Thus each 
point is translated through a distance Vr 2 +s 2 and in a direction 
parallel to the directed line from (0, 0) to (—r, —s). 
In the case of T, we rotate the new axes about the origin 
clockwise through angle 6 so that they now coincide with 
the initial axes. Then any point (x, y) is moved to a new point 
(x', y') by a clockwise rotation about the origin through angle 
6. By solving the equations of T, we get 
x =x cos 0+y sin 6, y' = —x sin 0+y cos 6.