384 M. Ampere's Memoir 



Lj Ly; (fig. 6) is found at one of the extremities of the first and 

 in the middle of the second, we obtain the momentum of rota- 

 tion resulting from the mutual action of these two conductors, 

 with the addition of those referring to each of the angles 

 L,OL", LyjOL", of which the two cotangents are equal and of 

 a contrary sign ; so that in marking the distances L,,L" and 

 L^L" by ?• and ?•', and the perpendiculars O P, O P' by p and 

 y, we have for that momentum 



^ it' \^p + p' + (?■' — r) cot s]. 

 Let us moreover suppose that the length O L" = a of the 

 conductor which has one of its extremities in O is equal to 

 half O L, or C) L^^ of the other, and let us call 9 the half P O L" 

 or POL,; of the angle L;;OL"= e, we shall find 



p = a cos S, J)' = a sin S,r = 2a sin fi, i' = 2 a cos S, 



_ 1— tang=(>_ l-cot»^_ 

 2 tang ^ 2 cot ^ ' 



the value of the momentum of rotation therefore is 



I • V f /I • /il— tanfir*^; . ^ 1— cot'^ ) 



^air \ cos 9 — sm 9 ^— + sm Q — cos 9 [ , 



^ ( tang i cot ^ ) 



or 



i aii' [cos 9 tang- 9 + sin 9 cot^ 9J = i a z /' [sin 9 tang 9 + cos 9 cot $]. 



It is sufficient to double the expression, suppressing the 

 denominator 2, for that produced by the action of the two 

 conductors L'L'', L^L,, of the same length, and the centres of 

 which are at the point O round which one of them is supposed 

 to be moveable. 



In the instrument of which I have just spoken, there are two 

 rectilinear conductors of equal length, moveable round their 

 centres ; from each of these centres, and sufficiently apart that 

 there may not be between the conductors a sensible mutual 

 action, project two other rectilinear conductors half the length 

 of the others ; these are fixed, and form between themselves 

 an angle that may be varied at will : the same electric current 

 runs through the six conductors ; so that in every one of the 

 fixed ones, and in that part of the corresponding moveable 

 conductor nearest to it, its cui'rent is in a contrary direction, in 

 order that the latter may keep in a steady equilibrium in the 

 perpendicular direction on the right line, which divides into 

 two equal parts the^angle of the two fixed conductors whose 

 action it experiences. As it is this latter angle which is given 

 immediately above the graduated arc attached to one of these 

 fixed conductors, it is desirable to introduce its half, which 

 we represent by *), instead of 9 in the expression of the mo- 

 mentum of rotation 



M s= I a » z' (sin fl tang 9 + cos 9 cot 9) 



which 



