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.

Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P **Appertain** To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.

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

X (xa) ki (xb) k2 (xc) k3...axibx2cx3...xx = (AB) hi (AC) h2 (BC) h3...A11 4 13 A1,14131 A B I ?C"' B C "' X (XA) ki (XB) k2 (XC) k3...AXB122cCk...X If this be of order e and **appertain** to an nie L Eke-/1+2m =e, h i+h2+ï¿½ï¿½ï¿½+221+ji+j2+ï¿½ï¿½ï¿½+kl+li =n, hi+h3+..ï¿½+222+ji+j3+ï¿½ï¿½ï¿½+k2+12 = n, h2+h3+ï¿½ï¿½ï¿½+223+j2+%3+ï¿½.ï¿½+k3+13 =n; viz., the symbols a, b, c,...

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

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.

Knox had from the first proclaimed that "the teinds (tithes of yearly fruits) by God's law do not **appertain** of necessity to the kirkmen."

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.