380 Permanent Magnetism [CH. xi 



On equating these two values for / we obtain an equation which may be 

 expressed in the form 



w)dS = (370), 



.(371), 



where the integration is over a closed surface bounded by 8 and 8', and 

 I, m, n are the direction-cosines of the outward normal to the surface at any 

 point. From equation (370), the necessity of condition (367) follows at once. 



Condition (367) is most easily proved to be sufficient by exhibiting an 

 actual solution of the problem when this condition is satisfied. We have to 

 shew that, subject to condition (367) being satisfied, there are functions 

 X } Y, Z such that 



dZ dY 



o -*- u 



oy dz 



dx dz__ 

 o o ^ 



dz dx 



- = u 



dx dy 



for if this is so, the required line integral is I (IX + mY + nZ) ds. 

 By inspection a solution of equations (371) is seen to be 



X=fvdz, Y=-iudz, Z = (372), 



for it is obvious that the first two equations are satisfied, and on substituting 

 in the third, we obtain 



dY dX (( du dv\, fdw, 



5 ^~ = ~~ o -5~ }dz = r^r dz = w, 



dx dy J\ dx dyj J dz 



shewing that the proposed solution satisfies all the conditions. 



442. The absence of symmetry from solution (372) suggests that this 

 solution is not the most general solution. The most general solution can, 

 however, be easily found. If we assume it to be 



-j 



udz+Y', Z = Z' ......... (373), 



then we find, on substitution in equations (371), that we must have 



aZ'_8F_' d_X^_^dZ^ dY' = dX' 

 dy dz ' dz dx ' dx dy 



and if we introduce a new variable % defined by % = X'dx, we find at once 

 that 



dx } dv' dz ' 



