may be called directions of free translation. Directions perpendicular to a plane of geomet- 

 rical symmetry for the surface of the body always have this property. As a general rule, if 

 the body moves in any other direction, the fluid exerts a torque upon it, and this torque must 

 be balanced by external forces if rotation is to be prevented. In cases of special symmetry 

 there may be many directions of free translation; and some bodies, such as a sphere, can move 

 freely in any direction. 



A force applied in a direction of free translation produces acceleration in that direction 

 only. A force applied in any other direction can be resolved into three perpendicular compo- 

 nents, each of which acts in a direction of free translation. Each component will then produce 

 a component of acceleration in its own direction, of the same magnitude as if the other com- 

 ponents of force were absent; and the total acceleration will be the vector sum of the three 

 component accelerations so produced. It the coefficients of inertia are different in the three 

 directions, the resultant vector force and the resultant vector acceleration will not be parallel. 

 As a simple example, to accelerate a massless thin disk through the fluid in a direction ob- 

 lique to its plane, the applied force must necessarily be perpendicular to the plane of the disk. 



To prevent rotational acceleration, it may be necessary also to apply a suitable torque. 



Besides pure translation, other types of steady motion not requiring the application 

 of external forces are possible. The most important case is that in which the surface of the 

 body has two planes of symmetry and the line of intersection of these planes passes through 

 the center of gravity of the body itself and is a principal axis of inertia for the body. Then a 

 steady rotation is possible about that axis; and a torque applied about such an axis generates 

 rotation about it in accord with the formula previously described. In special cases several or 

 many such axes of free rotation may exist. 



Two-dimensional motion may be further complicated by the presence of circulation about 

 the body, which then necessarily has the form of an infinite cylinder. In translational motion 

 the circulation gives rise to the familiar transverse force or lift; and the presence of circula- 

 tion may make steady rotation of the cylinder impossible in the absence of external forces. 

 Otherwise the statements that have been made for the three-dimensional case held also for 

 two-dimensional motion. 



In any case, the forces required to produce a given acceleration, translational or rota- 

 tional, are independent of the motion already existing and are the same as if the fluid were 

 at rest. This is easily seen from the pressure equation, as stated in Equation [9e]. Accel- 

 eration of the fluid motion is equivalent, at any time <j, to the superposition upon the flow 

 already existing at that time of an incremental flow that starts from rest. Since this added 

 flow does not alter the velocities as they exist at time <j, its only effect on the pressure at 

 time t^ is to add to the value of d(f)/dt a term that depends upon the acceleration but not on 

 the existing motion. It may happen that part of the total acceleration is actually due to hydro- 

 dynamical forces brought into play by the motion of the body through the fluid, such as the 

 forces that have just been described; then the additional acceleration produced by the external 

 forces is the same as it would be if these hydrodynamical accelerations were absent. 



382 



