256 OF THE EQUILIBRIUM OF FLOATATION, 



For since g is the centre of gravity 

 of the immersed part of the body, or of 

 the fluid ADB, it is also the centre of all 

 the forces or weights of the particles of 

 fluid in ADB, tending downwards; but 

 because the body is at rest, the same 

 point g is also the centre of all the 



pressures of the fluid tending upwards, by which the weight of the 

 body ADB H is sustained in a state of equilibrium. 



Now, it is manifest, that the sum of all the downward forces, is 

 equal and opposite to the sum of all the upward forces, otherwise the 

 body could not be in a state of rest ; but the direction in which the 

 weight of the body tends downwards from c, is perpendicular to the 

 horizon; consequently, the line CD which passes through G and g, 

 the centres of gravity of the whole body and of the part immersed, 

 must also be perpendicular to the horizon ; for if it is not, the body 

 must have a rotatory motion, but according to the hypothesis, the body 

 is at rest; therefore, the line CD is perpendicular to the horizon. 



From Propositions III. and VI., it is obvious, that for a floating 

 body to remain at rest, or in a state of equilibrium, two conditions 

 must obtain, and these are, 



The weight of the floating body, and that of the fluid dis- 

 placed, must be equal to one another. 



This is manifest from the inference under the third Proposition, and 

 the second condition to which we have alluded forms the substance of 

 Proposition VI., viz. 



The centre of gravity of the whole body and that of the 

 part immersed, or of the fluid displaced, occur in the same 

 vertical line. 



It is extremely obvious, from the nature of the subject, that both the 

 above conditions must have place ; for if the first do not obtain, the 

 body will ascend or descend in a direction which is perpendicular to 

 the horizon ; and if the second fail, the body will turn about its centre 

 of gravity, until the centre and that of the fluid displaced occur in 

 the same vertical line ; and if both these conditions fail together, the 

 body will partake of a progressive and a rotatory motion at one and 

 the same time. 



From the Proposition which we have demonstrated above, two or 

 three very useful inferences may be deduced, as follows. 



