# Algebraically Sentence Examples

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.He solved quadratic equations both geometrically and

**algebraically**, and also equations of the form x 2 "+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.It may be shown

**algebraically**that under theseconditions the n roots of the above equation in r2 are all real and positive.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.Further, the numerator factor establishes that these are not all

**algebraically**independent,, but are connected by a syzygy of degree order 6, 6.By Carnot's principle, in all irreversible processes, dH/0 must be

**algebraically**less than do, otherwise it would be possible to devise a cycle more efficient than a reversible cycle.AdvertisementOne more covariant is requisite to make an

**algebraically**complete set.The general equation of degree 5 cannot be solved

**algebraically**, but the roots can be expressed by means of elliptic modular functions.