100 APPLICATION OF THE PRELIMINARY RESULTS 



In iron and other similar bodies, g is very nearly equal to 1, 

 and therefore the directive force in the interior of a hollow sphe- 

 rical shell is greatly diminished, except when its thickness is 

 very small compared with its radius, in which case, as is evident 

 from the formula, it approaches towards the original value /, 

 and becomes equal to it when this thickness is infinitely small. 



To give an example of the use of the second formula, let it 

 be proposed to determine the total action upon a point p, situate 

 on one side of an infinitely extended plate of uniform thickness, 

 when another point P, containing a unit of positive fluid, is 

 placed on the other side of the same plate considering it as a 

 perfect conductor of magnetism. For this, let fall the perpen- 

 dicular PQ upon the side of the plate next P, on PQ prolonged, 

 demit the perpendicular pq, and make PQ = 5, Pq = u, pq = v, 

 and t = the thickness of the plate ; then, since its action is evi- 

 dently equal to that of an infinite sphere of the same thickness, 

 whose centre is upon the line QP at an infinite distance from P, 

 we shall have the required value of the total potential function 

 at p by supposing a t = a + 1, a infinite, and the line PQ pro- 

 longed to be the axis from which the angle 6 is measured. 

 Now in the present case 



TT_ _J_ __ \ _ _ V 77 (i)*.-*- 1 

 " ~ 2 -2r(a- b) cos + (a - 6) 2 } 



and the value of the potential function, as before determined, is 



From the first expression we see that the general term 

 U r -+-* i s a quantity of the order (a - I} 1 r~ l ~ l . Moreover, by 

 substituting for r its value in u, 



neglecting such quantities as are of the order - compared with 

 those retained. The general term CT/'V"*' 1 , and consequently 

 Z7, (i) , ought therefore to be considered as functions of - = 7. 



