the Galvanic Current through Iron. 145 



unstable equilibrium which would then of necessity prevail in 

 certain parts of the wire, very unlikely ; moreover the value 

 of the integral, so far as it comes into consideration, is inde- 

 pendent of these, (b) The differences of direction of the mo- 

 lecules will quickly diminish with M increasing, consequently 

 with diminishing deviation from the axis of the wire ; that is, 

 the integral which, divided by the difference of its limits, re- 

 presents that mean value will be included within limits rapidly 

 approaching one another, and at values of M which are great 

 in comparison with D can be replaced by a differential, and 

 consequently that quotient by a differential quotient ; the final 

 value of it then represents the extreme value of the work of the 

 current for saturated longitudinal magnetism. 



The work is, in the second case, greater than in the first, 

 from two causes : — first, because the work increases with the 

 force to be overcome ; an understanding of this can be obtained, 

 without a knowledge of those general expressions, in the fol- 

 lowing manner: we have 



^A_ 2D + Hsin(fr D(D + Hsin(ft) 2 

 3D W + W 3 



or 



BA_ D + Hsinft DH 2 cos 2 (ft 

 BD -1 W + W 3 



Now W can be written in the form 



W=\/(D + Hsin</>) 2 + H 2 cos 2 0. 

 Hence the sum to be subtracted from 1 in the expression for 

 ^-p. is greatest just when 



D + H sin </> = H cos <f> ; 

 and in this case it is equal to 



Thence it follows that ^— - is never <0. (We can also bring 

 dA . o u 8 



^-p. into the form 



(D + H sin c/>) 3 + 2H 2 D cos 2 <f> + H 3 sin <j> cos 2 </> 



w ; ' 



from which we bring out the same conclusion by the develop- 

 ment of W.) 



Bat what has here been proved for f-rv holds also for - — . 



