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

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592.]
MAGNETIC INDUCTION.
215

rectangle, are y0 and z − ½dz. The corresponding value of G is


(8)

and the part of the value of p which arises from the side A is approximately


(9)
Similarly, for B,
For C,
For D,


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


(10)


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


(A)


and consider these as the constituents of a new vector , then, by Theorem IV, Art. 24, we may express the line-integral of , round any circuit in the form of the surface-integral of , over a surface bounded by the circuit, thus


(11)
or (12)

where ε is the angle between and ds, and η that between and the normal to dS, whose direction-cosines are l, m, n, and , denote the numerical values of and

Comparing this result with equation (3), it is evident that the quantity I in that equation is equal to , or the resolved part of normal to dS.

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