Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/247

 Rh rectangle, are y0 and z − ½dz. The corresponding value of G is and the part of the value of p which arises from the side A is approximately

Adding these four quantities, we find the value of p for the rectangle

If we now assume three new quantities, a, b, c, such that

and consider these as the constituents of a new vector $$\mathfrak{B}$$, then, by Theorem IV, Art. 24, we may express the line-integral of $$\mathfrak{A}$$, round any circuit in the form of the surface-integral of $$\mathfrak{B}$$, over a surface bounded by the circuit, thus where ε is the angle between $$\mathfrak{A}$$ and ds, and η that between $$\mathfrak{B}$$ and the normal to dS, whose direction-cosines are l, m, n, and $$T\mathfrak{A}$$, $$T\mathfrak{B}$$ denote the numerical values of $$\mathfrak{A}$$ and $$\mathfrak{B}$$

Comparing this result with equation (3), it is evident that the quantity I in that equation is equal to $$\mathfrak{B} \cos\eta$$, or the resolved part of $$\mathfrak{B}$$ normal to dS.

592.] We have already seen (Arts. 490, 541) that, according to Faraday s theory, the phenomena of electromagnetic force and