Two spherical soap bubble coalesce. If `nu` be the change in volume of the contained air, A is the change in total surface area then show that 3PV + 4AT = 0 where T is the surface tension and P is atmospheric pressure.

Explanation

To solve the problem, we need to analyze the situation of two spherical soap bubbles coalescing. Let's break down the steps systematically.

### Step 1: Understand the Initial Conditions
When two soap bubbles coalesce, they form a new bubble. Let the radii of the two initial bubbles be \( r_1 \) and \( r_2 \). The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volumes of the two bubbles are:
...