of floating Bodies , and the Stability of Ships. 81 
that given angle of inclination ; for by solving the equation 
we obtain n = — =±= v/ -- 1 -~ ■ - — • Thus, if 
2 V 12 — 1 2S X 
12 n — I2»‘ — 2 
it should be required to ascertain the specific gravity which 
will cause the solid to float in equilibrio at the angular dis- 
tance of 23 0 29' from the upright, we 
have 
0.26000, and the specific gravity required, that is, w = .5 - j- 
.26 = .76, or n = .5 — .26 = .24. Thus we find from this 
calculation that there are two specific gravities which will 
cause the solid to float in a position of equilibrium at the same 
angular distance 23 0 29' from the original situation with a flat 
surface horizontal ; a conclusion which it is easy to verify by 
substituting .76 for n in the equation ~ == s*: the re- 
sult is that V = — - jj -f f -, the same as in the former instance, when 
n was assumed = .24. 
In the application of analytical investigation to the solution 
of problems, it is always necessary to keep distinctly in view the 
conditions on which the investigation has been founded ; for 
however correct the solution may otherwise have been, any in- * 
advertence in this respect will unavoidably lead to error and 
inconsistency. The investigation by which the floating position 
of the solid is determined after it has changed its position from 
an equilibrium of instability, when one of the flat surfaces was 
parallel to the horizon, has proceeded on a supposition that the 
surface of the fluid intersects the parallel surfaces YH, WV, 
(fig. 6.) in the points R and Z; but if the two surfaces inter- 
sected by the fluid should be the inclined sides HV, VW, or 
in other words, if the point of intersection Z should be si- 
tuated between H and V, neither the geometrical construction 
MDccxcvr. M 
