294 BELL SYSTEM TECHNICAL JOURNAL 



--, \dU ,„ \dU ,„ \dU ... 



vVx = -^-, ^Vy = -^- > ^^ = ~ "3~ • (9) 



r dx r dy r dz 



Like U, it is a function of x, y, z not only explicitly, but also implicitly 

 through r; we must therefore discriminate with care between total 

 and partial derivatives. For reference, here are the formulae ^ con- 

 necting derivatives of the one type with those of the other: 



d d , dr d d , X d d , , ^ d /.^n 



T" = ^ + T~ ^ = ^ + ~ ^ = T" + cos (x, r) — , (10 a) 



ax dx dx dr dx r dr dx dr 



d d dr d d y d d / \ ^ nr. x,\ 



:7- = T- + ^ 3- = V- + T- = v- + cos (y, r) — , (10 6) 



dy dy dy dr dy r dr dy dr 



d d , dr d d , z d d , , . d ,._ . 



:r = T" + -H" T" = T" + ~ "^ = -^ + cos (2, r) — , (10 c) 



dz dz dz dr dz r dr dz dr 



d _ d 5x6 dy d dz d 

 dr dr dr dx dr dy dr dz 



= ^ + cos (r, x) — + cos {r, y)^ + cos {r, z) —. {\Q d) 



The procedure consists in forming the expression for the true diver- 

 gence of W, to wit: 



A- w dW..dW,.dW^ .... 



^^"^ = -^ + -^+-^' (1^) 



and integrating it over the volume comprised between two surfaces: 

 outwardly, the surface S over which the values of U are preassigned, 

 and which envelops the origin at which the value of 5 is to be com- 

 puted; and inwardly, an infinitesimal sphere centred at the origin. 



It will turn out that the volume-integrals of the various terms either 

 vanish, or else may be converted into area-integrals over the two sur- 

 faces. Now the area-integral of any function / over the surface of a 

 sphere of radius R may be written as 



A = ^-kF}]] (12) 



in which/ stands for the mean value of/ over that surface — a statement 



^ In deriving the first three of these formulae, use the relation r^ = x^ -h ^ + z- in 

 evaluating drjdx, drjdy, drjdz. In deriving the last, remember that in forming a 

 derivative with respect to r at a point P, the increment dr is always measured along 

 the line extending from the origin through P, for which line x/r = cos {x, r) = const.; 

 ylr — cos {y, r) = const.; z/r = cos (z, r) = c(,n-;t.; hence dx/dr = cos (.v, r), etc. 

 Or one may arrive by geometrical intuition at the formula, 



d/dr = cos (r, x)d/dx + cos (r, y)d/dy + cos (r, z)djdz, 



from which (10 d) may be obtained by means of (10 a, b, c). 



