158 Intelligence and Miscellaneous Articles, 



This magnet may be regarded as consisting of an infinity of mag- 

 netic elements of constant and infinitely little thickness. The 

 quantity of magnetism T developed in each of the elements varies 

 with its distance a> from one of the extremities of the bar, and may 

 be represented by Y=<^(V). 



The quantity of free magnetism upon the element dx of the bar 

 is the difference of the values of T at the points which have for 

 abscissae m and x-\- dx; so that the quantity of free magnetism at 

 the point situated at the distance x from the extremity of the bar 



dY 



is y— — =z<p\x). The quantity of free magnetism in one point 



is therefore proportional to the quantity of magnetism of the bar. 



Supposing the bar composed of two symmetrical parts, let us 

 consider one of them in particular. Starting from the extremity, 

 the free magnetism diminishes till at a certain distance A it becomes 

 sensibly nil ; beyond, the function retains a sensibly constant value 



¥»• ' 



The pole of this portion of the bar is the centre of a system of 



parallel forces proportional to the quantities of free magnetism ; 

 the distance x from the pole to the extremity of the bar is deter- 

 mined by the theorem of the moments, 



X J ydx=z J xydx. 







If we suppose the bar sufficiently long, so that the magnetism 

 developed at the extremity may be neglected, we find easily 



\ 

 X0(X)=ty(X)- f <t>(x)dx. 



The magnetizing has given rise to attractive forces between the 

 several magnetic elements, or, according to Ampere's theory, be- 

 tween the parallel currents circulating in the solenoid formed by 

 the magnet. The element whose abscissa is x is solicited by forces 

 exerted by the elements around it, forces proportional to the quan- 

 tities of magnetism of the acting elements, and of which the in- 

 tensity rapidly decreases in proportion as the distance augments. 



The increment of the virial, relative to the point under consi- 

 deration, which results from the magnetizing may be represented by 

 ju0(V), — n denoting a function of the distance, being at the same 

 time proportional to the quantity of magnetism developed in the 

 bar, and consequently to the free magnetism. Besides, ty(x) is a 

 function proportional to the quantity of free magnetism of the bar ; 

 consequently every term /i0(#) of the virial is proportional to the 

 square of that quantity. 



Designating by I half the length of the bar, the increment of the 

 virial which results from the magnetizing is, for the half of the bar, 



c l r x C l 



J Hf(x)dx=^ }x$(x)dx+) P<p(x)dx. 



