66 LECTURE IX. 



the loss of any force. But unless some art has been previously employed 

 in producing a partial separation, it will frequently be found that the stick 

 has strength enough to break the glasses before it gives way. 



When an insulated body revolves round an axis in any direction, the 

 state of revolution cannot be permanent, unless the axis be so situated that 

 the centrifugal forces on each side of it balance each other.* It is obvious 

 that this must happen in a homogeneous sphere, whatever may be the 

 situation of the axis ; and it has been demonstrated, that when the body 

 is of an irregular form, there are at least three different axes, situated at 

 right angles to each other, round which the body may revolve in an equi- 

 librium either stable or tottering. It may also be shown that if a body, 

 revolving round any axis, receive at the same time an impulse whicli 

 would cause it to revolve round a second axis in another direction, the two 

 revolutions will be combined, and will form a single revolution round a 

 third axis in an intermediate position, which will remain at rest until it be 

 displaced by some new force, provided that it be one of the axes of permanent 

 revolution : so that no body can revolve round a moveable axis without a 

 continual disturbing force. And when an irregular body begins to move on 

 an axis incapable of equilibrium, its revolution will be gradually altered, 

 so as to approach continually to a revolution round one of the natural 

 axes ; but it will never pass beyond the state of equilibrium, as in many 

 other cases of deviation from such a state ; since the momentum pro- 

 duced by the excess of centrifugal force in one part of the revolution is 

 destroyed in another. For a similar reason, if a stick be thrown, in a 

 horizontal position, with a rotatory motion, it will fall in the same position 

 much more certainly than if it were thrown without any rotation ; for 

 any small disturbing force, which might be sufficient to turn it into a verti- 

 cal position during the course of its path, will only produce, when com- 

 bined with the rotatory motion, a slight change of the direction of the rota- 

 tion, which will confine the deviation of the stick from a horizontal posi- 

 tion within narrow limits. 



The subject of preponderance, or of the action of w r eights or forces coun- 

 teracted by other forces and incumbered with foreign matter to be put in 

 motion, requires for its discussion a previous knowledge of the simple 

 operation of forces, of the conditions of equilibrium, and of the estimation 

 of rotatory power. The consideration of the effects of preponderance 

 enables us to determine, in some circumstances, the best possible propor- 

 tions of the powers of machines for producing the required effects in the 

 most advantageous manner. For, in order that motion may be produced, it 

 is not sufficient that there be an equilibrium, in procuring which a part only 

 of the power is expended, but there must be an excess of force above that 

 which would be necessary for the equilibrium; and it is often of con- 

 sequence to know what portion of the power must be employed in each 

 w r ay, in order that the greatest effect may be produced in a given time. 

 We are sometimes told, that what we gain in power we lose in time. t In 



* Segner, de Motu Turbinum, Halle, 1755, first pointed out the three natural 

 axes of rotation of all bodies. Their existence was demonstrated by Eujer in 1760. 

 See Hist, et Mem. de 1'Acad. 1758, p. 154; 1760, p. 176. 



