A ship saslling from sea into a river sinks X mm and on discharging the corgo rises Y mm. on proceeding again into sea the ship rises the Zmm. Find te specific gravity of sea-water assuming the faces of ship are vertically along the line of sea water.

Answer

To solve the problem of finding the specific gravity of seawater given the conditions of the ship's movement between the sea and the river, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Ship**:
- When the ship is floating, the weight of the ship (W_s) plus the weight of the cargo (W_c) is balanced by the buoyant force (B).
- The buoyant force is given by the formula:
\[ B = \text{density of fluid} \times \text{volume submerged} \times g \]
- The volume submerged can be expressed as the area of the ship's base (A) multiplied by the height submerged (H_s).

2. **Establish the Initial Condition in Seawater**:
- Let the density of seawater be \(\rho_s\). The equilibrium condition when the ship is in seawater can be expressed as:
\[ W_s + W_c = \rho_s \cdot A \cdot H_s \cdot g \quad \text{(Equation 1)} \]

3. **Condition When the Ship Moves into the River**:
- When the ship moves from seawater to river water (density \(\rho_r\)), it sinks an additional \(X\) mm. Thus, the new submerged height is \(H_s + X\).
- The equilibrium condition in the river becomes:
\[ W_s + W_c = \rho_r \cdot A \cdot (H_s + X) \cdot g \quad \text{(Equation 2)} \]

4. **Condition After Discharging Cargo**:
- When the cargo is discharged, the ship rises by \(Y\) mm. The new submerged height is \(H_s + X - Y\).
- The equilibrium condition after discharging the cargo in the river is:
\[ W_s = \rho_r \cdot A \cdot (H_s + X - Y) \cdot g \quad \text{(Equation 3)} \]

5. **Condition When the Ship Returns to the Sea**:
- Upon returning to seawater, the ship rises by \(Z\) mm. The new submerged height is \(H_s + X - Y - Z\).
- The equilibrium condition in seawater is:
\[ W_s = \rho_s \cdot A \cdot (H_s + X - Y - Z) \cdot g \quad \text{(Equation 4)} \]

6. **Subtracting Equations**:
- Subtract Equation 2 from Equation 3 to eliminate \(W_s\):
\[ 0 = \rho_r \cdot A \cdot (H_s + X - Y) \cdot g - \rho_r \cdot A \cdot (H_s + X) \cdot g \]
- This simplifies to:
\[ W_c = \rho_r \cdot A \cdot Y \cdot g \quad \text{(Equation 5)} \]

7. **Using Equation 4**:
- Substitute Equation 5 into Equation 4:
\[ \rho_r \cdot A \cdot Y \cdot g = \rho_s \cdot A \cdot (H_s + X - Y - Z) \cdot g \]
- Cancel \(g\) and \(A\):
\[ \rho_r \cdot Y = \rho_s \cdot (H_s + X - Y - Z) \]

8. **Finding Specific Gravity**:
- Rearranging gives:
\[ \rho_s = \frac{\rho_r \cdot Y}{H_s + X - Y - Z} \]
- The specific gravity of seawater is defined as:
\[ \text{Specific Gravity} = \frac{\rho_s}{\rho_r} \]
- Substituting for \(\rho_s\):
\[ \text{Specific Gravity} = \frac{Y}{Y + Z - X} \]

### Final Answer:
The specific gravity of seawater is given by:
\[ \text{Specific Gravity} = \frac{Y}{Y + Z - X} \]