For this reason the umbrae A1, A 2 are said to be **contragredient** to xi, x 2.

Are said to be **contragredient** when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z ï¿½..., Z = A 3 X +ï¿½3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.

Are **contragredient** with the d- variables x, y, z, ...

If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities **contragredient** to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the **contragredient** line variables-it is completely determinate if its leading coefficient be known.