This equation is generally true for any series of transformations, provided that we regard H and W as representing the algebraic sums of all the quantities of heat supplied to, and of work done by the body, heat taken from the body or work done on the body being reckoned negative in the summation.

For further developments of the theory of determinants see Algebraic Forms. (A.

If we define the positive direction along a tube of electric force as the direction in which a small body charged with positive electricity would tend to move, we can summarize the above facts in a simple form by saying that, if we have any closed surface described in any manner in an electric field, the excess of the number of unit tubes which leave the surface over those which enter it is equal to 47r-times the algebraic sum of all the electricity included within the surface.

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As an algebraic 1 It appears from Joly's and Macfarlane's references that J.

The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

For an algebraic solution the invariants must fulfil certain conditions.

Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra.

Energyequations, such as the above, may be operated with precisely as if they were algebraic equations, a property which is of great advantage in calculation.

Laws of Algebraic Form.

This fact is of extreme importance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities.