378 Permanent Magnetism [CH. xi 



/ a , after replacing x, y, z by x + 8x, y + fry, 4- &z, will be / fe . Hence if we 

 denote this new value of I a by I a + /, we shall have 



so that I a -I b = - 



and the value of this quantity can be obtained by the ordinary rules of the 

 calculus of variations. 



We have 

 8 



rB fJrf, rS J rS A 



I X3?<fof SX~ds + X~(Bx)ds 

 } A ds J A ds J A ds^ 



dX, dX , \ dx , [ v *l s [*dX , 

 &y+ &z \-j-ds +\XSx\ - -j- Sxds, 

 dy dz ) ds \_ \ A } A ds 



and since &c vanishes both at A and B, the term X8x may be omitted, 

 and the whole expression put equal to 



Xs dX s dX.\dx fdXdx^dXdy dXdz\ , ) , 

 -5 oo! -f -5 By + ^ Sir ) -3 [ -5- -j- + -5- -/ 4- 5 :r) &&> cfe. 

 ^ 9y ^ J ds \dx ds dy ds dz dsj j 



or again, on simplifying, to 



(*fiX (* dx dy\ dX f' dz dx\] , 



\-~- (By -j- -%x-f }--^- (Sx-r -fa-r H ds. 

 JA(dy\ ds ds) dz \ ds ds] } 



This may be written in the form 



jf*||? (Bydx - Bxdy) - d ~ (Bxdz - Szdx)l ............ (364). 



FIG. 112. 



Now in fig. 112, let P, Q, P f be the points x,y,z\ x + dx, y + dy, z + dz\ 

 and x + Sx, y + %, z + z. Let dS denote the area of the parallelogram 

 PQQ'P', and let I, m, n be the direction-cosines of the normal to its plane. 

 Then the projection of the parallelogram on the plane of xy will be of area 



