28 KESEAKCHES ON II. 



well with each other when made in identical circumstances and with equal lengths 

 of total resistance, but disagreed when the resistances were considerably different. 

 I knew not to what cause to attribute these irregularities, whether to inexactness 

 in the principles of Ohm, or to some unknown cause inherent in the experiments 

 themselves; but the formulae will show that it arose at least in part from the 

 smaller deviation of the needle in one case than in the other. The breadth of the 

 bundle of wires had also a certain influence that could not be conveniently appre- 

 ciated without the use of the formulas. But these things will be understood better 

 after a complete exposition of the experiments, which will be given very soon. 



We have considered until now a simple circular current, but the case of Mr. 

 Plana, of a globe surrounded by currents passing through the poles, is very different 

 from this ; it can, however, be reduced to it with certain restrictions. 



Let us suppose now only two circular currents, disposed around a globe, passing 

 through the same poles, and disposed symmetrically on both sides of the magnetic 

 meridian. It may be shown that, when the needle is very small in comparison with 

 the diameter of the globe, the ratio of their action at different points of the polar 

 diameter is equal to that of a single circular current, whose intensity is equal to 

 the sum of both, and which is in the meridian plane. For, these currents making 

 two equal angles + d^, — d^ with the meridian, we have from formula (7), § prec, 

 for the sum of their actions on the needle deviating D degrees,^ 



Tsin. D+ Tsin.D = Z cos. {D — d') + Z cos. {D + d'). 



When the needle is very small, in comparison with the circles, and the angle D 

 and d^ are not very great, the values of Z differ but little from the mean value 

 Zm; therefore 



2 Tsin. D = Z {cos. [D — d") + cos. {D + d^)) = 2 Zm cos. D cos. d\ 

 that is 



T tang. D = Zm cos. d^ = hf {D,!P) cos. d': 



This formula, for another position of the needle, becomes 

 T tang. D = kif {Dpi,) cos. d> ; 

 and when D is the same, we shall have 



l- ^/(Aj>) whence A =-^1^^). 



These results are confirmed by experiment, as we shall see. 



§ 8. Description of the Instkuments used to verify the FoRMcrL.^. 



The apparatus required for these experiments consists of: — 



1. A simple voltaic electromotor or battery of constant force. 



2. One or two multiplicator galvanometers. 



3. A rheostat, with its accessories. 



4. A circle or globe, around which wires are coiled, adjustable on its stand. 



* The last term in the second member of equation (7) being omitted, as very small, in the experiments. 



