Next, suppose that the formula of transformation of a quadratic differential form is known, e.g.,
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(5)
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then taking two independent differentials dx, dy, δx, δy, and writing
in place of dx, dy, we get the formula of transformation of the bilinear form
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(6)
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We may multiply this equation by itself, multiplying both sets of differentials according to Grassmann's rule. The resulting equation is
This gives
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(7)
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or
Multiplying (6) by (1) according to Grassmann's rule, we obtain
![{\displaystyle {\begin{array}{l}\left[(Eb-Fa)\delta x-(Ga-Fb)\delta y\right]dx\ dy\\\qquad =\left[(E'b'-F'a')\delta x'-(G'a'-F'b')\delta y'\right]dx'\ dy'.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8702178a37da2b241c2080933ab8b7736a5cb0)
This gives the formula of transformation of a linear form
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(8)
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which may be called the reciprocal of the first.
Multiplying this equation by
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respectively, we get
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