II. ELECTRICAL RHEOMETRY. 23 



distance is := +^ Z sin. d or i^ L cos. d, as the case may be, we shall have its 

 momentum, which tends to turn the needle round its centre of motion in that 

 direction. 



We shall find the equation, first for the case of the two first systems of forces. 



The terrestrial component parallel to F acts in the direction of this axis, while, 

 on the contrary, that parallel to OZ acts against it; the condition of equilibrium 

 will then be — supposing both poles equally strong — 



(3) L sin. d (2 Tcos. d' + {Ya + Yb)) — L cos. d {[Za + Zb) — 2 T sin. d') = 0. 

 If in this we substitute the value of Ya, Yb, from (1) and (2), making 



Q^ = ^^cos.a{2E'-2WF'-c'E'), 

 If ¥ c^ 



we shall have 



Ya + Yb = Qa' {L sin. d — Z^) — Qh" {L sin. d + Z,) 

 = ( ga^ — Qb') L sin. d—{Qa'+ Qb') Z^ . 

 Substituting this value in (3), the equation of equilibrium of the declinating needle 

 will be reduced to 



(4 ) 2 Tdn. {d+d') = {Za^ Zb) cos. d— ( Qa^ — Qb')L sin? d + ( Qci} + Q¥) Z^ sin. d. 

 If NS (Fig. 7) represent the orthogonal projection of the magnetic meridian, 

 i)^that of the plane of the circular current, and ^5 that of the needle on the 

 horizontal plane YOZ, we shall have 



NCD = d\ ACD = d. 

 J'ormula (4) can be applied to the dipping needle, provided the plane of the cur- 

 rents be perpendicular to that of rotation ; to the meridian line, SN, we must then 

 substitute the line of inclination of the needle according to the latitude of the place, 

 and that equation in such a case will be sufiicient. 



In the case of the declination needle suspended not by an axis, but on a pivot, 

 the system of forces parallel to OX, combined first with those parallel to Y, and 

 then with those parallel to OZ, will give two other equations from which the incli- 

 nation of the needle will be determined ; but as these equations are of very little 

 use in practice, and can easily be found after what we have said of equation (4), 

 we shall omit them. 



Formula (4) is reduced to a very simple one when the needle has its middle on 

 the diameter of the circle J, because then 



Za = Zb, Qa' = — Qb\ Z^ = 0. 

 Therefore it becomes 



(5) Tsin. {d + d') = Z cos. d— Q^ L sin.^ d, 

 Z being a function of L, d, p, R. 



If, besides this, the needle be concentric with the circle, and the plane of the 

 currents coincide with the meridian, then we have, d^ = 0, and Za = Zb, and the 

 formula will be reduced to this very simple one. 



(6) T tang. d^= Z — (^ L sin. d tang, d, 

 where d is the deviation of the needle produced by the current. 



