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The image you provided contains formulas related to fluid dynamics and the study of viscosity, specifically dealing with Stoke's law and parameters related to fluid resistance and the motion of particles in fluids.

Let’s break down the two main parts of the formula and solve for the questions posed:

### 1. Formula for Density of Particle ( \(\bar{\rho}_1\) ):
\[
\bar{\rho}_1 = \frac{m}{\frac{1}{6}\pi d^3}
\]
This represents the density \(\bar{\rho}_1\) of a spherical particle, where:
- \(m\) is the mass of the particle,
- \(d\) is the diameter of the particle.

### 2. Formula for Viscosity (\(\eta\)):
\[
\eta = \frac{1}{18} \cdot \frac{(\rho_1 - \rho) \cdot d^2 \cdot g \cdot \tau}{L \cdot \left(1 + 2.4 \frac{d}{D}\right)}
\]
Where:
- \(\eta\) is the viscosity of the fluid,
- \(\rho_1\) is the density of the particle,
- \(\rho\) is the density of the fluid,
- \(d\) is the diameter of the particle,
- \(g\) is the gravitational acceleration,
- \(\tau\) is the time (possibly related to the settling time of a particle),
- \(L\) is the length,
- \(D\) is another dimension, likely the diameter of the container or tube where the experiment is conducted.

### The questions asked:
#### 1. \(\frac{\Delta \rho_1}{\rho_1}\):
This asks for the relative change in the particle density. To calculate this, we would need to know the change \(\Delta \rho_1\) and the original density \(\rho_1\). The formula could be written as:

\[
\frac{\Delta \rho_1}{\rho_1} = \frac{\rho_1' - \rho_1}{\rho_1}
\]
Where \(\rho_1'\) represents the new density of the particle after some change.

#### 2. \(\frac{\Delta \eta}{\eta}\):
This asks for the relative change in the viscosity. Similar to the first question, the formula would be:

\[
\frac{\Delta \eta}{\eta} = \frac{\eta' - \eta}{\eta}
\]
Where \(\eta'\) is the new viscosity of the fluid after some change.

### To proceed further with the calculations or to explain in more detail, we would need the specific changes (\(\Delta \rho_1\) and \(\Delta \eta\)) or more context on the variables involved.

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Would you like details on how to approach these changes in density and viscosity based on experimental data?

Here are some additional relative questions based on this:

1. How is the diameter \(d\) of a particle typically measured in these experiments?
2. How does the viscosity \(\eta\) affect the settling time \(\tau\) of particles in a fluid?
3. What role does gravitational acceleration \(g\) play in determining particle settling in a fluid?
4. Can the relative change \(\Delta \eta / \eta\) be used to predict flow behavior in different fluids?
5. How does changing the container diameter \(D\) influence the viscosity calculation?

**Tip:** When calculating relative changes, it's important to differentiate between absolute and relative values, especially in complex systems like fluid mechanics.

The image contains formulas related to fluid dynamics and the study of viscosity, particularly Stoke's law, involving changes in density and viscosity.

Explore how to calculate relative changes in viscosity and density using Stoke's law. This detailed problem involves fluid dynamics concepts, including particle density and viscosity formulas. Ideal for undergraduate students studying fluid mechanics.

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