PEIRCE. — DERIVATIVE OF A SCALAR POINT FUNCTION. 347 



orthogonal coordinates which have been published since Lamp's clas- 

 sical treatise 2 appeared. 



Given a function, u, it is, however, not generally possible to find a 

 system of orthogonal functions of which u shall be one, and it is often 

 convenient for a physicist to differentiate a physical function, U, with 

 respect to another, u, without considering the existence of any other 

 related functions. A physical point function has a value at every point 

 in space which is not altered by changing the system of coordinates 

 which fix the position of the point, and it is well to define the deriva- 

 tive of U with regard to u in a manner which shall emphasize the fact 

 that the derivative is an invariant of a change of coordinates and which 

 shall not assume that two functions (v, w) can be found orthogonal to 

 each other and to u. When U and u are considered by themselves and 

 not regarded as coordinated of necessity with other similar quantities, 

 it is usually, if not always, the case that a " normal " derivative 3 is 

 required. 



The normal derivative, at any point, P, of the differentiable scalar 

 point function U, with respect to the differentiable scalar point function 

 u, may be defined as the limit, when PP' approaches zero, of the ratio 



U P > -Up 



— -, (20) 



Up' — up J 



where P' is a point so chosen on the normal at P of the surface of con- 

 stant u which passes through P, that up' — up shall be positive. If 

 ( U, u) denotes the angle between the directions in which U and u in- 

 crease most rapidly, the normal derivatives of U with respect to u and 

 of u with respect to U may be written 



^•cos(£7, u) and ^■co&(U i ii\ (21) 



n u hu 



If hu= hu these derivatives are equal. An example of this is the 

 equality of dn j dr and dr / dn in a familiar application of Green's 

 Theorem, where n and r represent the normal distance from a given 

 surface and the distance from a given fixed point respectively. If U 

 and u happen to be expressed in terms of a set (x, y, z) of orthogonal 



2 Lame, Lecons sur les Coordonnees Curvilignes et leur Diverses Appli- 

 cations; Salvert, Memoire sur 1'Emploi des Coordonnees Curvilignes; Dar- 

 boux, Lemons sur les Systemes Orthogonaux et les Coordonnees Curvilignes; 

 Goursat, Cours d'Analyse Math6matique. 



3 Peirce, Short Table of Integrals, Theory of the Newtonian Potential 

 Function ; Generalized Space Differentiation of the Second Order. 



