ON PHYSICAL LINES OP FOKCE. 469 



governor for steam-engines *, we find idle wheels whose centres are capable of 

 motion. In all these cases the motion of the centre is the half sum of the 

 motions of the circumferences of the wheels between which it is placed. Let 

 us examine the relations which must subsist between the motions of our vortices 

 and those of the layer of particles interposed as idle wheels between them. 



PROP. IV. To determine the motion of a layer of particles separating tw 

 vortices. 



Let the circumferential velocity of a vortex, multiplied by the three direc- 

 tion-cosines of its axis respectively, be a, ^8, y, as in Prop. II. Let I, m, n be 

 the direction-cosines of the normal to any part of the surface of this vortex, 

 the outside of the surface being regarded positive. Then the components of the 

 velocity of the particles of the vortex at this part of its surface will be 



n/3 my parallel to x, 



ly na parallel to y, 



ma lft parallel to z. 



If this portion of the surface be in contact with another vortex whose velocities 

 are a, ft, 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 u is the velocity of the particles in 

 the direction of x, 



u = lm(y'-y)-ln(f?-p) ........................ (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 x in unit of time. 



2 , 



Let x t , y lt z 1 be the co-ordinates of the centre of the first vortex, 2 , y 2 

 those of the second, and so on. Let V lt F 2 , &c. be the volumes of the first, 

 second, &c. vortices, and V the sum of their volumes. Let dS be an element 

 of the surface separating the first and second vortices, and x, y, z its co-ordinates. 

 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 



* See Goodeve's Elements of Mechanism, p. 118. 



