Let these projections be denoted **algebraically** by x1, xi,.

To form a conception of this problem it is to be noted that since the position of the body in space can be computed from the six elements of the orbit at any time we may ideally conceive the coordinates of the body to be **algebraically** expressed as functions of the six elements and of the time.

Dodgson periodically published mathematical works - An Elementary Treatise on Determinants (1867); Euclid, Book V., proved **Algebraically** (1874); Euclid and his Modern Rivals (1879), the work on which his reputation as a mathematician largely rests; and Curiosa Mathematica (1888).

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.

Thus, to divide i by i +x **algebraically**, we may write it in the form I+o.x+o.x 2 +o.x 3 +o.x 4, and we then obtain I I +0.x+0.x2+0.x3 '+0.x4 = I' x+x2 - x 3 + x4 I+x I+x' where the successive terms of the quotient are obtained by a process which is purely formal.