344 VIII. NEWTONIAN POTENTIAL FUNCTION. 



116. Second Differential Parameter. If for the function U 

 we take a constant, say 1, 



du du du 



-fa~Ty == W = > ^ ==0 > 

 and we have simply 



23) - jyp r cos (P v ri) dS = fj d ~^dS 



The function 



which, following the usage of the majority of writers, we shall denote 

 by z/F', was termed by Lame 1 ) the second differential parameter of V. 

 As it is a scalar quantity it will be sufficiently distinguished from 

 the first parameter if we call it the scalar parameter. We have 

 accordingly the theorem giving the relation between the two: - 



The volume integral of the scalar differential parameter of a 

 uniform continuous point -function throughout any volume is equal 

 to the surface integral of the vector parameter resolved along the 

 outward normal to the surface S bounding the volume. 



We may obtain a geometrical notion of the significance of z/F 

 in a number of ways. In the neighborhood of a point 0, let us 

 develop V by Taylor's theorem, calling the coordinates of neighboring 

 points with respect to 0, x y y, z, then 



where the suffix denotes the value at 0. 



Integrating the value of V -- F throughout the volume of a 

 small sphere with center at 0, we have 



<).///*+! <SX///"-+ i 



+ 



1) Gr. Lame. Legons sur les Coordonnees curvilignes et lews diverses Applica- 

 tions. Paris, 1859, p. 6. 



