26 RESEARCHES ON H. 



a single element when the current is a simple circular line, whose plane passes 

 through the middle of the needle, and the length of the needle can he considered as 

 infinitely small But in practice, all these conditions cannot be realized, since, using 

 a single current, the effect is exceedingly small, and the errors of position have a 

 great influence on the results. On the contrary, using a bundle of wires, in order 

 to have a stronger effect, the problem is completely different, since each current 

 cannot lie mathematically in the plane passing through the middle of the needle. 

 The influence of all these circumstances was unknown to me when I made the first 

 series of experiments ; now, however, we can appreciate their influence and discover 

 the causes of the discrepancies between the experiments and the numbers given by 

 Mr. Plana. 



Let us first consider the case of a single current, after which we shall consider 

 that of a globe. Since the formula which expresses the intensities of the forces by 

 means of the deviation of the needle is known, it will be very easy to verify by 

 experiment Mr. Plana's problem. To do so, it will be sufficient to settle the needle 

 in different points of the vertical diameter OX, and measure its deviation. 



In fact, supposing the circle in the plane of the magnetic meridian, the formula 

 (6) for a position p, and a deviation d, may be expressed by 



T tan. d = hf [p, d), 

 and for another position by 



T tan. c?i = hf {Pi, <^i); 

 whence h being = h, 



(m) ^^^^- ^ ^ fJP^d) ^ 

 tan. di f {pi, di) ' 



the deviations being observed, and / calculated, this equation must be satisfied. K 



this happens, in fact, the formula is proved to be exact. 



"When I made the above experiments, I did not know these formulae, and of 

 course was obliged to have recourse to another expedient. I considered, also, that 

 the measure of deviation could not be accurate enough in this case, because I was 

 obliged to use small graduated circles, the construction of the machine not allowing 

 of larger ones. K large circles could have been used, the want of formulae would 

 have been supplied by experimental tables of proportional degrees of force, as is 

 done in Melloni's experiments, it being very easy to construct good ones. Another 

 way seemed therefore preferable, which might be independent of the variations of 

 angular deviation, and of every analytical expression of forces. It is the following : 



"Whatever may be the action of a current on a needle, it shall always be 

 expressed by 



h being the intensity of the current, and / a function dependent on the position of 

 the needle relative to the circular current. If we change the place of the needle, 

 and from the centre transfer it to another position, for instance, at /oths of the 

 radius on the vertical axis OX, the new force acting will be 



Now these two forces can be made equal to one another by varying the intensity 

 of the current, and they will be equal to each other when they are in equihbrium 



