284 Prof. Maxwell on the Theory of Molecular Vortices 
nB—my parallel to 2, 
ly—na_ parallel to y, 
ma—lB parallel to z. 
If this portion of the surface be in contact with another vortex 
whose velocities are «', 6’, y', 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 w is the velocity of the particles in the direction of a, 
u= tm(y!—y)—4n(8'-8), «ss (27) 
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 2 in unit of 
time. 
Let 2), yj, 2; be the coordinates of the centre of the first vor- 
teX, Xo) Yo) Zo those of the second, and so on. Let V,, Vo, &e. 
be the volumes of the first, second, &c. vortices, and V the sum 
of their volumes. Let dS be an element of the surface separa- 
ting the first and second vortices, and z, y, z its coordinates. 
Let p be the quantity of particles on every unit of surface. 
Then if p be the whole quantity of particles transferred across 
unit of area in unit of time in the direction of a, the whole mo- 
mentum parallel to « of the particles within the space whose 
volume is V will be Vp, and we shall have 
Vor Spd S ei nude - wake een 
the summation being extended to every surface separating any 
two vortices within the volume V. 
Let us consider the surface separating the first and second 
vortices. Let an element of this surface be dS, and let its 
direction-cosines be /,,m,,n, with respect to the first vortex, and 
1,, Mg, 2. With respect to the second; then we know that 
1+1,=0, m+m,=0, m+7,=0. . . (29) 
The values of a, 8, y vary with the position of the centre of 
the vortex; so that we may write 
da da de 
Oy = ty F dé (%—2) + dy (%a—41) + de (2g—%,), + (80) 
with similar equations for 8 and y. 
The value of wu may be written :— 
