the Stability of Ships. 
a 33 
appears that v/sine O x sine P 
V' i — tang . 1 i F x tang . 1 S # 
sec. 4 F X sec. S 
which quantity being substituted instead of \/ sine O x sine P, 
in the value of FW just found, the result will be 
FW == FE x/i — tang. 2 \ F x tang. 2 S. 
It is found also, from trigonometrical rules, that 
* WO = FE X and since 
^ V i — tang. 2, - F x tang . 2 S 
WQ : WM : : sine S : rad. we have 
WM 
FE x tang. 2, 4 F x tang. S xsec. S 
sine S x ^ i — tang . 1 j- i' x tang . 1 S 
•, or because 
tang. S 
sine S 
sec. S, WM = FE x 
tang . 1 1 F xsec . 1 S 
a/ i — tang . 1 1 F xtang , 1 S 
and, since 
FW= FE x -s/ 1 —tang.* i F x tang.* S, and FM=WF+WM, 
we obtain the value of FM = FE x 
sec . 1 4 F 
V i — tang . 1 |Fx tang 1 S 
Therefore ME = FM - FE = FE x 
sec/ 
V i — tang . 1 j F x tang . 1 S 
1 , 
sec . 1 4 F 
1 . 
and ET = FE x sine S x — =. 
V i — tang . 1 \ F x tang . 1 S 
This value of ET is inferred from supposing the area BFA to 
1 2 , 
represent the entire volume immersed, and which = — 
1 4 X tang. \ F 
t being equal to the line BA. 
If, the sides BC, AH, remaining the same, the figure and 
magnitude of the immersed volume should be changed, so as to 
be represented by any other quantity V-f-, the line ET will be 
increased or diminished in the inverse proportion of the en- 
tire volumes immersed, that is 
as V : 
e 
sec . 1 4 F 
x „ : : FE x sine S x — 
4 x tang. 2 F %/ i — tang . 1 4 F xtang . 1 S 
t 
l : ET, 
And, since FE 
3 tang. 4 F ’ 
* See Appendix. 
MDCCXCVIIL 
f See pages 213 and 214. 
Hh 
