/ 



on a iiexi; Electro-dyiiamic Experimetit. 377 



from the equations of the chxiiit, so that we may choose these 

 equations in such a mannei* that the integral 



cos» /3 



is not I'educed to zero between the Umits. In order to remove 

 the total of the tangential force, it is necessary that the co- 

 efficient of this iiitegral be null; which gives the relation 

 sought for, 2 k + 71 — I — 0. 



In oi'der to form a j aster idea of the integi'al 



d r. 



/ 



we may conceive, round the middle of the element d s taken for 

 a centre, an infinity of spherical surfaces, which divide the com- 

 plete circuit into infinitely small arcs, so that the two extreme 

 spherical surfaces touch it at the two points of this circuit, which 

 are, one the furthestfrom, and the other the nearest to, the mid- 

 dle of the element ; then we may consider the complete circuit 

 as being composed of two branches terminating at these two 

 points, and both divided into an equal number of infinitely 

 small arcs, so that every arc of one branch corresponds with 

 that of the other branch comprised between the two same 

 consecutive spherical surfaces: for two corresponding arcs we 

 have then the same value of r, and the values of d ?• are equal, 

 but of contrary signs, for the current cannot go, in withdraw- 

 ing from the element d s into one of the branches, without 

 going, in approachuig it, into the other. Thence we see why 

 the integral //(r) dr is always null when it is taken in the 



whole extent of the complete circuit, since this integral is then 

 composed of elements which are, two by two, of equal value, 

 but of different signs. 



It would be the same with /J{r) cos- /3 d r, if cos- /3 had the 

 same value for any two corresponding elements ; ex. gr. if 

 these two elements were always situated symmetrically on the 

 two sides of a plane raised perpendicularly on the middle of 

 d s'; but if, on the contrary, in one of the two branches the 

 absolute value of cos /3 for every element is greater than for 



its correspondent, //"(r) cos /3 d r will be composed of two 

 series of terms, one of wiiich will contain only positive terms* 

 and the other negative terms ; so that each of the former shall 

 have an absolute value greater or smaller than that of the ne- 

 gative term corresponding witli it in the other series. Then 

 this integral can never be null ; and in order to make the 

 tangential force comformable to experience, we must have 

 2 k + n — \ = 0. 



Vol. 6(i. No. 331. iVw. 1825. SB Setting 



