38 _ Sequel of M. Ampéere’s Memoir 
around its centre, oscillates from side to side of its situation 
of equilibrium, when it is submitted to the action of two fixed 
conductors, each of which has one of its extremities at this 
centre, and whose length is a. In the instrument which I 
have contrived for verifying this result of my formula, it is not 
only these two conductors which act on that which is move- 
able, but also the circular portion of the voltaic circuit which 
joins the two other extremities of the fixed conductors: as the 
action which results from this portion is exerted in a contrary 
direction, a momentum is obtained of which the sign is opposed 
to that of the momentum of which we have just obtained the 
value, it must be added to the first; and what is very remark- 
able, the total momentum takes aform much more simple. In 
short, in naming M! the momentum of rotation produced by 
this arc, that which must be added to 2 — dé 
is evidently 2 —~ dé; 
as the radius of the arc s! is equal to a, we have s! = 2a6+C, 
ds/ 
whence dj = 
d ™’ dM’ 
dé 
But the tangential force in the direction of the element ds! 
and, consequently, 2 
r Isat qaqa cost B 
being 277 ds'd ——, 
and its momentum of causing this element to turn round its cen- 
tre being equal, and of a sign contrary to that whose value 
we are seeking, we have 
d2 M’ 54 cos? B 
— f— 1 Idol | a 
aay dsds'= — fZaids'd —_, 
aM’ » y (008? B" 2 
whence a 1s! =_— tail (= P me ait B as. 
id r 
Observing that it is necessary to integrate in the same man- 
ner in relation to the direction of the current as for recti- 
linear fixed conductors, we find 
cos 6! = — cos 6, 7’ = 2 asin 6, cos 6" = sin 4, r'= 2a cos 6, 
thus 
dM’ «+ /Cos? é sin? 6 1 
ee ee eT fobs) sind ee 
ds’ 4 cS cos 6 =g!u (cos 6 sin 6)(Sat 1), 
mee +1)dé. 
sin 4 cos 6 
dM’ ar 4 
and 4077 44 = aii! (cosé — sin 6) ( 
Uniting this momentum with that which we have called 
dM 
2--dé, 
