TERRESTRIAL MAGNETISM. 191 



6. 



If two points in space P°, P', be connected by an arbitrary 

 line, of which d s represent an indeterminate element, and if, as 

 before, 6 signify the angle between d s and the direction of the 

 magnetic force there existing, and -\Jr its intensity, then 



/Vcos e.ds= V - V^ 



if we extend the integration through the whole Hne, and desig- 

 nate by F°, V, the values of V at the extremities. 



The following corollaries of this fruitful proposition deserve 

 especial notice : — 



I. The integral / •>|r cos, 6 .ds preserves the same value by 

 whatever path we proceed from P° to P'. 



II. The integral / -^fr cos .ds, extended through the whole 



length of any re-entering cur\T, is always = 0. 



III. In a re-entering cun^e, if 6 is not throughout = 90°, a 

 part of the values of 6 must be greater and a part must be less 

 than 90°. 



7. 



Those points of space in which V has a value greater than V^, 

 are divided from those in which the value of V is less than V^, 

 by a surface in all the points of which F has one determinate 

 value = F°*. 



It foUows from the proposition in Art. V,, that in each point 



of this surface the magnetic force has a direction perpendicular 



to the surface, and towards the side where the higher values of 



V are found. Let (/ s be an infinitely small line perpendicular 



to the surface, and V^ + dV^ the value of V at its other extre- 



d F" 

 mity ; then the intensity of the magnetic force will be = — j — • 



The series of points for which V = V^ + d V^, form a second 

 surface infinitely near to the first, and at diflferent points in the 

 whole intervening space the intensity of the magnetic force is in 

 the inverse ratio of the distance apart of the two surfaces. 



• If the function V could have any arbitrarily chosen form, then in parti- 

 cular cases a maximum or a minimum value of V might correspond to an in- 

 sulated point, or to an insulated line, around which only greater or only less 

 values might be found, or it might correspond to a surface on both sides of which 

 there might be greater or on both less values. But the conditions to which the 

 function V is subjected do not allow the occurrence of such excepted cases. A 

 full development of this subject, as it is unnecessary for our jiresent object, 

 must be reserved for anolher occasion. 



N 2 



