There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

The possible causes of this widespread tendency of the mean age of a western community to increase appertain to the subject of the movement of the population, which is dealt with below.

Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.

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The functions of the king are those that appertain everywhere to the sovereign of a constitutional state.

This blending of the two systems of education produced the happy result of fitting this Moslem chief in an eminent degree both for the sacerdotal functions which appertain to his spiritual position, and for those social duties of a great and enlightened leader which he was called upon to discharge by virtue of that position.

There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.