31, 32] LAMELLAR VECTORS. 89 



near level surfaces, we see that, the numerator being constant, the 

 derivative in the direction of the normal, that is the value of the 

 vector parameter R, is inversely proportional to the normal distance 

 between the two infinitely near level surfaces of the function (p. 



Such a pair of surfaces will be called a thin level sheet or lamina. 

 For this reason a vector point -function that may be represented 

 everywhere in a certain region as the vector parameter of a scalar 

 point -function will be called a laminar, or lamellar vector (Maxwell). 



The scalar function cp (or its negative) will sometimes be termed 

 the potential of the vector E. 



32. Motion of the Center of Mass. Suppose that a system 

 of particles is under the influence only of forces acting between the 

 particles and depending on their mutual distances, and that the 

 constraints, if there be any, are such as to permit of a virtual dis- 

 placement which is represented by equal vectors for all the particles. 

 Then in our equation of d'Alembert's principle (18) let us put each 

 dx r equal to the same quantity A, each dy r equal to [i, and each dz r 

 equal to v. 



Supposing the system to be conservative, and using equations 23) 

 we have 



Now as the forces depend only on the mutual distances of the 



particles, and therefore only on the differences of their coordinates, 

 if we put 



ii = x l - x n , ^ =y 1 y n , $1 = *! **, 



n l=X n -i X n , 1? w _i=# n _i 



we shall have 



U=U(^,^,^.. 

 Accordingly 



l^l^HL i l?lk . ._LI ... = 

 dx : di dx l ~"~ d| 2 cx^ <?% ^^ n 



and likewise 



