346 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Sometimes s and s' are fixed directions so that I, m, n, I', m' , n', are 

 constants throughout T, and in this case the coefficients of du/dx, 

 du/dy, du/dz in (12) and (13) vanish. The mutual potential energy W, 

 of two magnetic elements, M, M f , of moments, m, m\ can be written 

 in the form 



'*£#} <M> 



m -m 



where r is the distance MM 1 and s, s' are the directions of the axes 

 of the elements. The force (due to the second magnet) which tends 

 to move the first magnet in the direction of its own axis is then 



— m-m , ( ) (17) 



2 as -as -as \r/ 



and these differentiations assume that the direction cosines of s and 

 s' are constants. 



In general, if s is the direction perpendicular to the level surface of 

 u, and if h is the scalar point function which gives the value of du/ds, 



dru (dh da dk du , dh da\ I, ,,. N 



FT = (a--5- + F--^- + 5;-r- ) k - ( 18 ) 



d$- \d.r d.r dy dg dz dz )\ 



In the case of oblique Cartesian coordinates in a plane, x increases 

 fastest in a direction which is not perpendicular to the line along 

 which it is constant. If the angle between the coordinate axes is w, 



— = h u - cos (.?•, h u ), — = h u ■ cos (y, h u ), — = h u ■ cos (s, h u ), 



du _ du sin (?/, s) du sin (,r, s) . . 



— — — r 1 h — • -. (1 J ) 



ds dx sm w dy sin w 



It is frequently necessary to differentiate one point function, U, with 

 respect to another, u, and the process usually appears in the form of a 

 kind of partial differentiation. If, for instance, U is to satisfy a differ- 

 ential equation in terms of a set of orthogonal curvilinear coordinates 

 of which u is one, the derivatives of U with respect to u are to be taken 

 on the assumption that the other coordinates remain constant. This 

 large subject has been treated exhaustively in the many works on 



