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ALGEBRAIC INVARIANTS 
PART I 
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ILLUSTRATIONS, GEOMETRICAL INTERPRETATIONS AND 
APPLICATIONS OF INVARIANTS AND COVARIANTS. 
1. Illustrations from Plane Analytics. If x and y are the 
coordinates of a point in a plane referred to rectangular axes, 
while x' and y' are the coordinates of the same point referred 
to axes obtained by rotating the former axes counter-clock 
wise through an angle 9, then 
T: x = x f cos 9—y' sin 9, y = x' sin 9-\-y r cos 9. 
Substituting these values into the linear function 
l = ax+by+c, 
we get a'x'+b'y'+c, where 
a' =a cos 9-\-b sin 9, b' — —a sin 9+b cos 9. 
It follows that 
a' 2 +b' 2 = a 2 +b 2 . 
Accordingly, a 2 -\-b 2 is called an invariant of l under every 
transformation of the type T. 
Similarly, under the transformation T let 
L=Ax+By+C=A'x'+B'y'+C, 
so that 
A' = A cos 9-j-B sin 9, B' = —A sin 9+B cos 9.