\(\therefore {P_1} = P + \frac{{4T}}{{{R_1}}}.\,\,\,{P_2} = P + \frac{{4T}}{{{R_2}}}\,\,and\,\,{P_3} = P + \frac{{4T}}{{{R_3}}}\)

\(Also\,{P_1}{V_1} + {P_2}{V_2} = {P_3}{V_3}\)

\(\therefore \left( {P + \frac{{4T}}{{{R_1}}}} \right)\frac{{4\pi }}{3}R_1^3 + \left( {P + \frac{{4T}}{{{R_2}}}} \right)\frac{{4\pi }}{3}R_2^3\)

\( = \left( {p + \frac{{4T}}{{{R_3}}}} \right)\frac{{4\pi }}{3}R_3^3\)

\(P\left( {\frac{{4\pi }}{3}R_1^3 + \frac{{4\pi }}{3}R_2^3 - \frac{{4\pi }}{3}R_3^2} \right)\)

\( + \frac{{4T}}{3}\left( {4\pi R_1^2 + 4\pi R_2^2 - 4\pi R_3^2} \right) = 0\)

\(P\left( {{V_1} + {V_2} - {V_3}} \right) + \frac{{4T}}{3}\left( {{S_1} - {S_2} - {S_3}} \right) = 0\)

\(PV + \frac{{4T}}{3}S = 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,3PV + 4ST = 0\)