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
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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