The periodic variations can be represented algebraically as the resultant of a series of harmonic motions in the following way: Let L be an angle which is increasing uniformly with the time, and let n be its rate of increase.

Further, the numerator factor establishes that these are not all algebraically independent,, but are connected by a syzygy of degree order 6, 6.

It may be shown algebraically that under theseconditions the n roots of the above equation in r2 are all real and positive.

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Forsyth has discussed the algebraically complete sets of ground forms of ternary and quaternary forms (see Amer.

The longitudes, latitudes and radii vectores of a planet, being algebraically expressed as the sum of an infinite periodic series of the kind we have been describing, it follows that the problem of finding their co-ordinates at any moment is solved by computing these expressions.

One more covariant is requisite to make an algebraically complete set.

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).

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