PEIRCE. — SPACE DIFFERENTIATION OF THE SECOND ORDER. 381 



Let the normal derivative,* at any point P, of a point function V, 

 taken with respect to another point function W, he the limit, as P Q ap- 

 proaches zero, of the ratio of V Q — V P to W Q — W P , where Q is a point 

 so chosen on the normal at P to the surface of constant W which passes 

 through P, that W Q — W P is positive: if, then, ( V, W) denotes the angle 

 between the directions in which V and W increase most rapidly, the 

 normal derivatives of Fwith respect to W, and of IF with respect to V, 

 may be written 



[D W V\ =h v -cos(V, W)jh m [DyW]=h w -co*(V i W)lhy: (11) 



if h v = h w , these derivatives are equal. 



With this notation (10) may be rewritten in the form 



2VO = A'[2M.] (12) 



If at any point Z) s 2 fl vanishes, it is easy to see from (10) that either 



the gradient (h 1 ) of /* vanishes at the point, or else the h and £2 surfaces 



cut each other there orthogonally. This latter case is exemplified in the 



familiar instance of the electrostatic field due to two long parallel straight 



wires of the same diameter, charged to equal and opposite potentials : if 



the wires cut the xy plane normally at P^ P 2 , and if the line joining 



these intersections be taken for x axis with the point midway between 



them for origin, the potential function is of the form V= A logr 1 /r 2 , 



where r x 2 = (x — a) 2 + if, r 2 2 = (x + a) 2 + y\ The intensity of the 



field, in absolute value, is h = 2 a Ajr x r 2 , and the second derivative of V 



taken along the line of force (that is, the rate at which the intensity of the 



— AaAx 



field changes) is numerically equal to — ^ % • 



r% ' r 2 



D s 2 V taken along a line of force vanishes, therefore, at all points on 



they axis, and at all such points the curve of constant V (i\ /r 2 = b) 



cuts the curves of constant h (r t r 2 = k) orthogonally. At points on the 



y axis the direction of the lines of force is parallel to the x axis, and the 



second derivative of Kwith respect to the fixed direction x happens to 



vanish here also where I = 1, yr- = 0, m = 0, -=- = 0. The quantity 



c)x ay 



h' does not vanish at any finite point. 



* Peirce, The Newtonian Potential Function, p. 116. A Short Table of Inte- 

 grals, p. 106. 



