86 APPLICATION OF THE PRELIMINARY RESULTS 



We have thus the means of comparing theory with experiment, 

 but these are details into which our limits will not permit us to 

 enter. 



The formula (b), which is strictly correct for an infinitely 

 small sphere, on the supposition of its magnetic particles being 

 arranged in a confused manner, will, in fact, form the basis of 

 our theory, and although the preceding analysis seems suffi- 

 ciently general and rigorous, it may not be amiss to give a 

 simpler proof of this particular case. Let, therefore, the origin 

 of the rectangular co-ordinates be placed at the centre of the 

 infinitely small sphere A, and OB be the direction of the parallel 

 forces acting upon it ; then, since the total quantity of magnetic 

 fluid in A is equal to zero, the value of the potential function V, 

 at the point p f , arising from A, must evidently be of the form 



T7_ ^ COS ^ 



-/*-; 



r representing as before the distance Op', and the angle formed 

 between the line Op , and another line OD fixed in A. If now 

 f be the magnitude of the force directed along OB, the constant 

 k will evidently be of the form k = a'f-, a being a constant 

 quantity. The value of F, just given, holds good for any 

 arrangement, regular or irregular, of the magnetic particles com- 

 posing A, but on the latter supposition, the value of F would 

 evidently remain unchanged, provided the sphere, and conse- 

 quently the line OD, revolved round OB as an axis, which 

 could not be the case unless OB and OD coincided. Hence 

 6 = angle BOp' and 



g/C080 



r 12 



Let now a, ft, 7, be the angles that the line Op = r makes with 

 the axes of x, y, z, and a', ft', 7', those which OB makes with 

 the same axes ; then, substituting for cos 6 its value 



cos a cos a' + cos ft cos ft' + cos 7 cos 7', 

 we have, since 



/cosa=JT, /cos/3=Y, 



v- a ' ^ cos a + ^ cos P 



