ON GENERALIZED SPACE DIFFERENTIATION OF 

 THE SECOND ORDER. 



By B. O. Peirce. 



Presented December 9, 1903. Received December 26, 1903. 



If one has to investigate the strength of a field of force defined by a 

 given scalar potential function, or to study the flow of electricity 

 in a massive conductor under given conditions, or to apply Green's 

 Theorem to given functions in the space bounded by a given closed 

 surface, or, indeed, to treat any one of a large number of problems in 

 Mathematical Physics or in Analysis, one often needs to find the numer- 

 ical value at a point, of the derivative of a point function taken in a 

 given direction. This has given rise to the familiar idea of simple space 

 differentiation and of the normal derivative of one scalar function with 

 respect to another ; indeed the properties of the first and of the higher 

 space derivatives of a function of n variables taken with respect to any 

 fixed direction in n dimensional space, have been treated very clearly 

 and exhaustively by Czuber.* 



It is sometimes desirable to use also the conception of general space 

 derivatives of the second order. This is the case, for instance, when 

 one is determining the rate of change of the intensity of a conservative 

 field of force at a point which is moving, either along a curved line of 

 force or on a curved surface related to such lines in a prescribed man- 

 ner. It is easy to define the general space derivative of any order of a 

 given function. 



This paper discusses very briefly a few elementary facts with regard 

 to generalized space differentiation of the second order, and treats first, 

 for the sake of simplicity, differentiation of functions of two variables, 

 in the plane of those variables. 



Plane Differentiation. 



Let there be in the xy plane two independent families of curves 

 (m = c, V ^ Z;) such that in the domain, R, one and only one curve of 



* E. Czuber, Wienerberichte, p. 1417 (1892). 



