PEIRCE. — DERIVATIVE OF A SCALAR POINT FUNCTION. 349 



the dielectric would be unchanged if /x were changed to O./u., where fi is 

 any scalar point function orthogonal to V. If the continuous dielectric 

 of a condenser in which the level surfaces of the inductivity, /j., are 

 equipotential be changed so as to make the new potential function 

 between the plates a function [ V =f ( V)] of the old, the new induc- 

 tivity must satisfy an equation of the form // = Q./x If ( V). If the V 

 and the /x surfaces are neither coincident nor orthogonal, V cannot be 

 harmonic, and if V is given and one value of the inductivity found, no 

 other value of the inductivity with the same level surfaces as this can 

 be found except by altering the old value at every point in a constant 

 ratio. If V does not satisfy Lamp's condition, a new value of the 

 inductivity found by multiplying the old value by any point function 

 orthogonal to V, will yield the same value of V, but the level surfaces 

 of the inductivity will be altered. If the V and the /jl surfaces are not 

 coincident, no change of the inductivity which leaves its surfaces un- 

 changed can make these surfaces equipotential. 



If a mass of fluid, the characteristic equation of which is of the form 

 V —f(p> T)i is at res t under the action of a conservative field of force 

 the components of which are X, Y, Z, 



dp dp _ dp 



— = p-2L } —=p-y, —= P -Z. (26) 



ox dy dz 



It follows immediately from these equations that p and V must be 

 colevel, and the normal derivative of p with respect to V shows that 

 equilibrium is impossible unless the distribution of temperature is such 

 that the equipotential surfaces are also isothermal. 



If the scalar point function, Jl 7 , is expressed in terms of the three 

 orthogonal point functions, u, v, w, the square of the gradient of W is 

 well known to be equal to 



^■m + ^y + ^( 



djvy 



dw J ' 



If the vector point function Q is expressed in terms of u, v, w, the 

 divergence of Q is equal to 



V w \_dii\h v -h w ) dv\h u 'hw) dw\h u -h v J J' 



If the normal derivatives of u and v with respect to w be denoted by 

 D w u and D w v, it follows from the definition that 



