the Stability of Ships. 
3°9 
Note to the Investigation , Page 233. 
\*Jf) -p-p v tang.* t F x tang. S x sec. S 
^ */ 1 — tang.* 4 F X tang.* S 
The investigation follows : 
FP : PO : : sin. O : sin. F ; wherefore OP = FP x ; or be- 
sin. O ’ 
cause FP*= FE x sec. \ F x > /'• 
sin. O 
sin. P 
, OP = FE 
FEx 
2 sin. f F jOP 
=, and — — PQ = FE x 
■v/sin.O x sin. P 
sec. 4 F x sin. F 
X v' sin. O x sin. P 
or 
sin. 4 F 
v' sin. O x sin. P 
^ 1 — tang.* 4 F x tang.* S PQ FE 
sec. 4 F x sec. S 
because v/sin. O x sin. P'f = 
tang. 4 F x sec. S 
X :r z.. 
v i — tang.* 4 F x tang. S 
ButPW: WF (=FExJ\/ 1 — tang. a |- F x tang. a S) :: sin.^-F 
: sin. P; therefore PW = FE x sin. {Fx x - ta - ng - ,s 
z sin. P 
and WO = PW— PO = FE x 
P£ x tang. 4 F x sec. S 
V \ — tang.* 4 F x tang.* S x sin. 4 F 
sin. P 
v" 1 — tang.* 4 F x tang.* S 
or 
WQ = FE x sin> i F — tang* 4 F x sin. 4 F x tang.* S — tang. 4 F x sec. S x sin. P 
sin, Px Vi — tang.* 4 F x tang.* S 
But sin. P = cos. S + \ F = cos.-|- F x cos. S — sin. *- F x sin. S, 
which being substituted for sin. P in the value of WQ just 
found, the result is 
WO — FE v sin -l F ~tang.*4F x tang*Sx sin.4F— sin.4F-p tang«4F x sin. 4 F x tang.S 
COS.4F x cos.S — sin.|F x sin.S x ^ 1 — tang ,* \ F x tang.*S 
or WO Fp v tang. 4 F x sin. 4 F x tang.S— tang.* 4 Fx tang.*S x sin. 4 F 
^ COS.4F x cos.S — sin.iF x sin.S x V'i — tang.* 4 F xtang.*S 
or WO FF V tang. 4 F x sin. | F x tang. S X 1 — tang. 4 F x tang. S 
^ cos.4 F X cos. S —sin. 4 F x sin. S x Vi — tang.*4F x 
* Demonstrated in page 232. 
f Page 308. 
tang.* 4 F x tang.* S 
t Page 233. 
or 
