82 Mr. Atwood’s Propositions determining the Positions 
nor the analytical investigation depending on it, can be applied, 
so as to ascertain the required position of equilibrium, a so- 
lution altogether different being required to determine the po- 
sition in which a solid under these conditions will float per- 
manently. It is, however, certain, that as long as the point of 
intersection Z is not lower than the point of the base H, the 
preceding solution will be applicable : it will be therefore ma- 
terial to find both the angle of inclination from the original 
position of unstable equilibrium, and the specific gravity of 
the solid when it floats permanently, with this condition an- 
nexed, i. e. that the surface of the fluid shall pass through one 
of the extremities of the base : the result of this solution will 
form a limiting value both of the angle of inclination and of 
the specific gravity, beyond which the preceding investigation 
not being applicable, another solution is required. 
Let AECD (Tab. IV. fig. 7. ),represent a vertical section of 
the square parallelopiped which rests permanently on the sur- 
face of the fluid IKDH, passing through the extremity of the 
base D. It is required to find the angle of inclination KDC 
from a position of equilibrium with a flat surface horizontal, 
and the specific gravity of the solid, when it floats in a state 
of equilibrium. Let the tangent of the required angle KDC 
be to radius as t to 1, and put CD = a ; let the specific 
gravity of the solid be to that of the fluid as » to 1. Then 
KC == at, and the area KCD == : and because as the area 
KCD is to the area AECD, so is n to 1, it follows that n = 
• l — ; and since by the preceding investigation* $*= ■ i * 
where s represents the sine of the angle of inclination from the 
* Page 79. 
