24 KESEAECHES ON II. 



If for the angle d, we use the absolute deviation D, from the magnetical meridian, 



in formula (5), it will become 



d^^ D — d} and 



(7) Tdn. D = Z cos. {D — d^) — Q" L sin? {D — d'). 



It is evident that these conclusions, and chiefly the (6), can be applied to an 

 elliptical conductor; and so formulae (4), (5), (6), express also the condition of 

 equilibrium in the case of the ellipse, supposing, however, j» = 0. 



Our principal object being, for the present, to compare with theory the experi- 

 ments of which we have spoken in the introduction, we shall not enter into further 

 details by continuing the discussion of the formulae. We may observe, however, 

 by the way, that the formula supposed to represent electric forces in the compass 

 of tangents, cannot be exact except by approximation, when the needle is exceed- 

 ingly short in comparison with the diameter of the circle. 



The value of the second member of equation (6) varies sensibly with the magni- 

 tude of the deviation, even for a needle 4 centim. long in a circle of 45 centim. 

 diameter, when the deviations differ only 5 or 6 degrees, as we shall shortly see. 



' The compass of tangents, however, may be used in the same way as the compass 

 of sines,^ and then it is exact for every length of needle. 



To prove this, it will be sufficient to cast a glance on the formulae (5), and (6), 

 because the value of -Z' — Q^ L sin. d tang, d when the needle is not concentric 

 with the circle may be expressed by 



hf{d,p), 

 and when it is concentric by 



Z — Q^ L sin. d tang, d -^^ Tcf {d). 

 To another intensity and deviation we have 



Z — Q^ L sin. di tang, d^ = \ f (di) . 



Therefore from the (6) 



/gx Tang , d _ h f {d) 



^ ' Tang.di hf{di)' 

 As / {d) differs from / (c?i) when the value of the length of the needle is sensible, so 



the factor of the second member L is never = 1 except when the needle is very short. 



/i 

 But using this instrument as the compass of sines used by Pouillet, in which the 

 plane of the currents is always kept passing through the axis of the needle, then 

 d^= 0, and the formula (5) gives 



Tsin. d^ = Z. 

 the second member also is always constant since d is always the same and = 0, 



therefore cos. d =^ 1 and 



.qx Sin, d' ^ JcfjO) ^ ^ 

 ^ '' Sin. d\ kj{0) h' 

 But we do not know how much we can trust to experiments made with this 

 last apparatus, because, after the needle has reached the maximum of deviation, the 



* For these two instruments, see Becquerel, Traits de I'Electr. et Mag. ; t. v. p. 262. 



