324 Hydrostatics. . 



but the tendency thus to return will gradually diminish as it 

 revolves, till after a time the body will incline the other way, 

 but before this tendency changes its sign (to borrow an expres- 

 sion from algebra), there will be a position in which it will be 

 nothing, and in which the body will neither incline to return to 

 its first position, nor to depart from it ; this, therefore, will be 

 its second position of equilibrium. Now we see that within this 

 part of its revolution, the body tends to return to its first posi- 

 tion, and consequently to depart from the second. Beyond this 

 point, the body tends to depart from its first position, and at the 

 same time from the second ; therefore the second position of 

 equilibrium is not stable, since on each side of it the body tends 

 to depart from it. Upon its passing this position, its tendency 

 to depart from it diminishes continally, till it becomes nothing ; 

 and beyond this the body tends to return toward its second 

 position. The point where this tendency is nothing, is a third 

 position of equilibrium, which is evidently stable ; for on each 

 side of it, the body tends to return to it, either approaching to- 

 ward or receding from its second position. If the third position 

 is stable, it may be shown by the same kind of reasoning that 

 the fourth is not, and that the fifth is, and so on. 



Thus, when the body returns to its first position, it will have 

 necessarily passed through an even number of positions of 

 equilibrium, alternately stable and unstable. 



441. It is important to be able to distinguish a stable position 

 of equilibrium in a floating body from one which is not so. In 

 Fi "14 or ^ er to ^ s ' ^ et us su PP ose a body which admits of being divid- 

 ed by a vertical plane HFI into two parts perfectly similar, both 

 as to form and density. Let us suppose, moreover, that this 

 body is made to depart from its position of equilibrium, in such 

 a manner that this section HFI remains vertical, and that after 

 having thus disturbed it, we leave it to itself without impressing 

 upon it any velocity; in this way the section HFI will remain 

 in the same vertical plane, during the whole motion of the body, 

 for the two portions being perfectly similar in all respects, there 

 is no reason why it should ever depart from the vertical plane in 

 which it was supposed to be first situated. For the same reason, 

 the centre of buoyancy will always be in the section HFI, as 



