This conflict arises not only from naturalization having been granted without the corresponding expatriation having been permitted, but also from the fact that birth on the soil was the leading determinant of nationality by feudal law, and still is so by the laws of England and the United States (jus soli), while the nationality of the father is its leading determinant in those countries which have accepted Roman principles of jurisprudence (jus sanguinis).

The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.

If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged.

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Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.

The expression in form of a determinant presents in general considerable difficulties.

We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

Decomposition of a Determinant into complementary Deter- minants.

Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities.

The minor Aik is aa, and is itself a determinant of order n-t.

We thus obtain for the product a determinant of order n.