198 ON FARADAY'S LINES OF FORCE. 



by that part of the whole effect at any point which is due to the distribution 

 of conducting media, and not directly to the presence of the sources. 



This quantity Q is rendered a minimum by one and only one value of p, 

 namely, that which satisfies the original equation. 



THEOREM V. 



If a, b, c be three functions of x, y, z satisfying the equation 



da db dc _ 

 dx dy dz~ 



it is always possible to find three functions a, ft, y which shall satisfy the equa- 



tions 



dfl _dy = 

 dz dy 



dy-^ = b 



dx dz 



_ 

 dy dx 



Let A = \cdy, where the integration is to be performed upon c considered 

 as a function of y, treating x and z as constants. Let B*=\adz, C = \bdx, 

 A' = $bdz, R = \cdx, C'*=\ady, integrated in the same way. 



Then 



will satisfy the given equations ; for 



dB dy [da , (dc , (db , (da 

 - = dz - dx - dx + 



and i 



dB dy (da , (da , (da 

 ' ~j ~T~ = I ~:r. dx + I -j- dy + | -y- 



= 0. 



