CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL 

 LABORATORY, HARVARD UNIVERSITY. 



THE CONCEPTION OF THE DERIVATIVE OF A SCALAR 

 POINT FUNCTION WITH RESPECT TO ANOTHER 

 SIMILAR FUNCTION. 



By B. Osgood Peirce. 



Presented December 8, 1909. Received January 5, 1910. 



In modern treatises on Mathematical Physics it is customary to de- 

 fine the derivative of a scalar function, taken at a given point in space 

 in a given direction, in a manner which emphasizes the fact that this 

 derivative is an invariant of a transformation of coordinates. Accord- 

 ing to this definition, 1 if through the point P a straight line be drawn 

 in a fixed direction (s), if on this line a point P' be taken near P so that 

 PP' has the direction s, and if u P , Up> be used to represent the values 

 at these points of the scalar point function u, then if the ratio 



?y — Up 

 PP' Kl) 



approaches a limit as P' approaches P, this limit is called the derivative 

 of u, at P, in the direction s. If u happens to be defined in terms of a 

 system of orthogonal Cartesian coordinates, x, y, z, and has continuous 

 derivatives with respect to these coordinates everywhere within a 

 certain region, the limit just mentioned exists in this region and its 

 value is 



— • cos (#, s) + — ■ cos (y, s) + °g- cos {z, s). (2) 



1 Hamilton, Elements of the Theory of Quaternions; Tait, Elementary 

 Treatise on Quaternions; Gibbs, Vector Analysis; Maxwell, Treatise on 

 Electricity and Magnetism; Webster, Dynamics of Particles and of Rigid, 

 Elastic, and Fluid Bodies; Jeans, Mathematical Theory of Electricity and 

 Magnetism; Lame, Lecons sur Ies Coordonn^es Curvilignes; Peirce, Theory 

 of the Newtonian Potential Function; Generalized Space Differentiation of 

 the Second Order; Czuber, Wienerberichte, 101 A , 1417 (1892); Boussinesq, 

 Cours d' Analyse Infinit^simale ; H. Weber, Die Partiellen Differential-Gleich- 

 ungen der Mathematischen Physik. 



