348 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Cartesian coordinates, the normal derivative of U with respect to u 

 can be written 



dll du dU du dU du 



B u u= ^' Fx+ \?y **'** (22) 



and it is easy to see that this is equal to the ratio of the derivatives of 

 U and u taken in the direction in which u increases most rapidly. 



It is occasionally instructive to use the conception of normal differ- 

 entiation in studying some of the general equations of Physics : thus 

 in the uncharged dielectric about an electric distribution, the potential 

 function, V, is connected with the inductivity of the medium, //., by the 

 familiar equation 



d ( ar\ a ( dV\ o / ev\ rt 



s("-*0 + *('-*) + 5V'--S7=* (23) 



in which /j. is to be regarded as a point function discontinuous in gen- 

 eral at each of a given set of surfaces at every point of which an equa- 

 tion of the form 



dV dV n 



*-S5 + "-ai = (24) 



is satisfied. Now (23) may be put into the form 



and according to Lamp's condition, the second term is a function of V 

 only, if the level surfaces of V are possible level surfaces of a harmonic 

 function. 



It is easy to make from (25), by inspection, such simple deductions 

 as those which follow in this paragraph. If V is harmonic, either the 

 dielectric is made up of homogeneous portions separated from one an- 

 other by equipotential surfaces, or the level surfaces of /* and of V are 

 everywhere perpendicular to each other. If V, though not harmonic, 

 satisfies Lamp's condition [V 2 ( V)/ h v 2 = F{ V)] the level surfaces of the 

 inductivity are equipotential ; and if the level surfaces of V and (x are 

 identical, V satisfies Lamp's condition. If when the plates of a con- 

 denser are kept at given potentials, the level surfaces of the inductivity 

 of the dielectric are equipotential, the value of the potential function in 



