The expression of the law in the Cartesian
coordinate
system.
i ds
m
dX
= 5
sin
T
a.
r
2
r t-i • t
1 U
Him
[Fig.]
dY
=
+
sin
r ß.
i
ds
7
2
ds
m
dz
=
,
sin
r
7
.
r
2
The direction of the force is
1
to
r
and to ds, hence
aa
+
bß
+
ci
=
0
a
'
a
+
b
*
ß +
c
'
7
=
0
furthermore a2
+
ß2
+
72
=1
hence,
be
1
-
cb
a
=
J
(öc1
-
cb
')2
+
(ca1
-
ac1)2
+
(ab'
-
öa '
)
2
ca'
-
ac1 ab
1
-
ba1
ß
=
7
=
J
J
cos
r
=
aa
*
+
bb
'
-fee'
sin
Y
J
(a2
+ fc2 +
c2
)
(a
' 2 +
5 '2
+
c'2)
-
(aa'
+
bbx
+
cc1)
=
J
(aö
1 -
ba1)2
+
(be
'
-
c£
'
)2 +
(ca1
-
ac
'
)
Hence
c/X
=
+
-
--(öc
1
-
cb
1
)
i
ds
m
, . x
dY
=
+
-
-j
(ca1
-
ac
'.
)
.^ dZ
i ds
m
. .
,
. ,
=
+
- »
(ab*,
-
ba
'
)
[Fig.]
The
diagram
shows that the
sign
we
have chosen is
correct
for
a
coordinate
system
of the kind indicated. To
simplify the formulas,
we
choose, in
analogy
with the treatment
of electrical
masses,
the south
magnetic
masses
with
negative signs.
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