ON PHYSICAL LINES OF FORCE. 479 



If the machine were suddenly stopped by stopping the driving wheel, each 

 wheel would receive an impulse equal and opposite to that which it received 

 when the machine was set in motion. 



This impulse may be calculated for any part of a system of mechanism, 

 and may be called the reduced momentum of the machine for that point. In 

 the varied motion of the machine, the actual force on any part arising from 

 the variation of motion may be found by differentiating the reduced momentum 

 with respect to the time, just as we have found that the electromotive force 

 may be deduced from the electrotonic state by the same process. 



Having found the relation between the velocities of the vortices and the 

 electromotive forces when the centres of the vortices are at rest, we must 

 extend our theory to the case of a fluid medium containing vortices, and 

 subject to all the varieties of fluid motion. If we fix our attention on any 

 one elementary portion of a fluid, we shall find that it not only travels from 

 one place to another, but also changes its form and position, so as to be elon- 

 gated in certain directions and compressed in others, and at the same time (in 

 the most general case) turned round by a displacement of rotation. 



These changes of form and position produce changes in the velocity of the 

 molecular vortices, which we must now examine. 



The alteration of form and position may always be reduced to three simple 

 extensions or compressions in the direction of three rectangular axes, together 

 with three angular rotations about any set of three axes. We shall first con- 

 sider the effect of three simple extensions or compressions. 



PROP. IX. To find the variations of a, ft, y in the parallelepiped x, y, z 

 when x becomes x + Bx; y, y + 8y; and z, z + Sz; the volume of the figure 

 remaining the same. 



By Prop. II. we find for the work done by the vortices against pressure, 



8 W=p$ (xyz) - - (a'yzSx + fizxSy + -/xy$z) (59) ; 



and by Prop. VI. we find for the variation of energy, 



(60). 



