on a new Electro-dynamic Experiment. 39 
aii’ (cosé — sin @) ages Aaiir/ 2sink» 
we have sin? @ cos? cos? 4 
dé, 
because, besides the equation sin# cos § = 4 cos 44 which we 
have deduced (page 394 of the former portion of this memoir) 
from the value of 6, §@=t2e=4 (+ —n)s 
we obtain also from this same value 
cos §— sing = V2 siniy. 
The action which causes the moveable conductor to oscillate 
is then proportionate to the sine of the quarter of the angle 
comprised between the directions of the two fixed rectilinear 
conductors, divided by the square of the cosine of the half of 
the same angle; it becomes null with this angle, as it ought to 
be, and infinite when they are directed following the same right 
r 
line, because then 1,=>: 
In the instrument intended for the measurement of these 
oscillations, the two extremities of the moveable conductor are 
also joined by a conductor forming a semi-circumference ; but 
account is only to be taken of the action exercised on its recti- 
linear portion; since the circuit formed by the two fixedre cti- 
linear conductors, and by the arc which joins the extremities 
of it, is a closed circle which cannot act on the circular por- 
tion of the moveable conductor. 
The value which we have found for the elementary momentum 
dM’ pa 2 Bil 2 pI 
yds = —haiay =" — Se tae 
ds’ 
expresses generally the action impressed by the little are d s! 
on a conductor of any form whatever, so as to make it turn 
round an axis elevated by the centre of this are perpendicularly 
to its plane: this action is then independent of the form of 
this conductor, and only depends on the situation of its two 
extremities relatively to the little are ds’; it is equal, as it 
ought to be, to the produce of the radius a by the value which 
we have obtained (see vol. Ixvi. p. 378) for the force which is 
exercised on the same moveable conductor by a small por- 
tion equal to ds! of a rectilinear conductor directed ac- 
cording to this are ds'. When we wish to see the action of an 
arc terminated, we must integrate afresh with relation to s', 
and this second integration generally gives a different result 
in the two cases; but this result is the same when the move- 
able conductor has one of its extremities in the axis, and the 
other on the circumference of which the arc s! makes a part. 
The only sign of the value which is obtained becomes changed, 
because 
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