Semisubmerged Ships 549 
The last inequality is Routh’s discriminant, which requires the real parts of all complex 
roots to be positive. 
At this point we must note that stability as used here refers to directional and depth 
stability, that is, depth stability means that after an arbitrary excursion in either z or 9, 
zy as a or S > @ 
and directional stability means 
ek essa(0) as t or S25 Coe 
This is to be distinguished from the case of the deep-operating submarine, where the 
body is considered stable as long as it does not go into a diverging trajectory but settles 
on a new path that is straight but not in the same direction as that prior to the disturbance. 
If we note that M ., Zy M gare small and that Zg = 0 (as may be proved for the doubly 
symmetric body under consideration) the coefficients of the characteristic equation take on 
the relatively simple form 
Bye | 
Wh M 
oO ew, toe, 
ee mM. ° ON 
2 y 
ele zh [M.Z,- M, + (m+ Z,)] - = 
Dimay, ne, (12) 
ite eee Mewhiy 7 ough 
- le (tt. - 1,2) 
2imenyz 
e = gar (42. - ¥,2,) 
omy, 
where M, =m — Z,;, the virtual mass in direction z, and N, =/ 
ment of inertia. 
y ~ Mg, the virtual mass mo- 
It can be noted that a, b, and c are the same as will be found in the equation of motion 
for the deeply operating body, with the exception of the term —Z,/M,. The quantity d is 
seen to be an effective damping coefficient that is arrived at by coupling the damping force 
and moment derivatives with the free-surface force and moment derivatives, and e is an effec- 
tive spring constant arrived at by a similar coupling of the static force and moment deriva- 
tives with Z, and M,. It is now necessary to say something about the characteristic of Z, 
and M, as functions of Froude number [12]. The force rates (Z, and M,) arise from the asym- 
metry of flow about the body when it is moving parallel to the water surface. At very low 
