64 DEFINITE INTEGRALS. [INT. III. 



34. Second Differential Parameter. 



If for the function U we take a constant, say 1, 



S=f =f => $& 



and we have simply 



The function 



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

 denote by AF, was termed by Lame* the second differential para- 

 meter 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 $ bounding the volume. 



We may obtain a geometrical notion of the significance of AF 

 in a number of ways. Applying the above theorem to the volume 

 enclosed by a small sphere of radius R, we have, since n is in the 

 direction of the radius, but drawn inwards, 



8F .. F S -FO 



-3- = lim -= !, 



v'R' jR=0 -^ 



where F is the value of F at the centre of the sphere, V s on the 

 surface. Now remembering the significance of a definite integral 

 as a mean, we have 



lim -^ {Mean of F on surface F at center} x Area of surface 

 R=o-tt 



= / 1 1 A Vdr = (Mean of A F in sphere) x Volume of sphere. 



* G. Lam6. Legons sur les Coordonnees cwvilignes et lews diverses Applica- 

 tions. Paris, 1859, p. 6. 



