284
Prof. Maxwell on the Theory of Molecular Vortices
If this portion of the surface be in contact with another vortex whose velocities are
, then a layer of very small particles placed between them will have a velocity which will be the mean of the superficial velocities of the vortices which they separate, so that if
is the velocity of the particles in the direction of
,
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since the normal to the second vortex is in the opposite direction to that of the first.
Prop. V.— To determine the whole amount of particles transferred across unit of area in the direction of
in unit of time.
Let
be the coordinates of the centre of the first vortex,
those of the second, and so on. Let
, &c. be the volumes of the first, second, &c. vortices, and
the sum of their volumes. Let
be an element of the surface separating the first and second vortices, and x, y, z its coordinates. Let
be the quantity of particles on every unit of surface. Then if
be the whole quantity of particles transferred across unit of area in unit of time in the direction of
, the whole momentum parallel to
of the particles within the space whose volume is
will be
, and we shall have
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the summation being extended to every surface separating any two vortices within the volume
.
Let us consider the surface separating the first and second vortices. Let an element of this surface be
, and let its direction-cosines be
with respect to the first vortex, and
with respect to the second; then we know that
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The values of
vary with the position of the centre of the vortex; so that we may write
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with similar equations for
and
.
The value of
may be written:—