Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/43

$W=-m_{2}{\frac {dV_{1}}{dh_{2}}}=m_{1}m_{2}{\frac {d^{2}}{dh_{1}dh_{2}}}\left({\frac {1}{r}}\right),$

(3)

$\quad ={\frac {m_{1}m_{2}}{r^{3}}}(\mu _{12}-3\lambda _{1}\lambda _{2}),$

(4)

where μ₁₂ is the cosine of the angle which the axes make with each other, and λ₁, λ₂ are the cosines of the angles which they make with r.

Let us next determine the moment of the couple with which the first magnet tends to turn the second round its centre.

Let us suppose the second magnet turned through an angle dφ in a plane perpendicular to a third axis h₃, then the work done against the magnetic forces will be ${\frac {dW}{d\varphi }}d\varphi$ and the moment of the magnet in this plane will be

$-{\frac {dW}{d\varphi }}=-{\frac {m_{1}m_{2}}{r^{3}}}\left({\frac {d\mu _{12}}{d\varphi }}-3\lambda _{1}{\frac {d\lambda _{2}}{d\varphi }}\right).$

(5)

The actual moment acting on the second magnet may therefore be considered as the resultant of two couples, of which the first acts in a plane parallel to the axes of both magnets, and tends to increase the angle between them with a force whose moment is

${\frac {m_{1}m_{2}}{r^{3}}}\sin(h_{1}h_{2}),$

(6)

while the second couple acts in the plane passing through r and the axis of the second magnet, and tends to diminish the angle between these directions with a force

${\frac {3m_{1}m_{2}}{r^{3}}}\cos(rh_{1})\sin(rh_{2}),$

(7)

where (rh₁), (rh₂), (h₁h₂) denote the angles between the lines r, h₁, h₂.

To determine the force acting on the second magnet in a direction parallel to a line h₃, we have to calculate

${\frac {dW}{dh_{3}}}=m_{1}m_{2}{\frac {d^{3}}{dh_{1}\,dh_{2}\,dh_{3}}}\left({\frac {1}{r}}\right),$

(8)

$=3{\frac {m_{1}m_{2}}{r^{4}}}\left(\lambda _{1}\mu _{23}+\lambda _{2}\mu _{31}+\lambda _{3}\mu _{12}-5\lambda _{1}\lambda _{2}\lambda _{3}\right),$

(9)

$=3\lambda _{3}{\frac {m_{1}m_{2}}{r^{4}}}\left(\mu _{12}-5\lambda _{1}\lambda _{2}\right)+3\mu _{13}{\frac {m_{1}m_{2}}{r^{4}}}\lambda _{2}+3\mu _{23}{\frac {m_{1}m_{2}}{r^{4}}}\lambda _{1}.$

(10)

If we suppose the actual force compounded of three forces, R, H₁ and H₂, in the directions of r, h₁ and h₂ respectively, then the force in the direction of h₃ is

$\lambda _{3}R+\mu _{13}H_{1}+\mu _{23}H_{2}.\,$

(11)