## 6 Conclusions
\begin{array}{c}\alpha HH(\psi+w)-(1+v)H(\psi)-(3+v)\alpha H(w)\\ +2(1+v)(\psi+w)+\\ +\varphi[-H(\psi)+H(w)+2(\psi+w)]=0.\end{array} \] (A33)  
Now, any regular function of cos\(\theta\) in the interval \(-1\leq\) cos\(\theta\leq 1\) may be expanded in a series of Legendre functions:  
\[ \begin{array}{c}P_{0}(\cos\theta)=1,\\ P_{1}(\cos\theta)=\cos\theta,\\ R_{2}(\cos\theta)=0.25(\cos 2\theta+1),\\ P_{n}(\cos\theta)=2\frac{1\times 3\times 5\times\ldots\times(2n-1)}{2^{n}n!} \times\\ \times\left[\cos n\theta+\frac{1}{1}\times\frac{n}{n-1}\cos(n-2)\theta\right.\\ \left.+\frac{1\times 3}{1\times 2}\times\frac{n(n-1)}{(2n-1)(2n-3)}\cos(n-4) \theta+\ldots\right]\\ \frac{d^{2}p_{n}}{d\theta^{2}}+\cot\theta\frac{d^{2}p_{n}}{d\theta^{2}}+n(n+1 )p_{n}=0,\end{array} \] (A34)  
\[H(P_{n})=-\lambda_{n}p_{n},\] (A35)  
\[HH(P_{n})=\lambda_{n}^{2}p_{n},\] (A36)  
in which \(\lambda_{n}=n(n+1)-2\), and \(n\) is an integer. Assuming the general expressions of \(\psi\) and \(w\) for any symmetrical buckling of spherical shells, we have:  
\[\psi=\sum_{n=0}^{\infty}A_{n}P_{n}\] (A37)  
\[w=\sum_{n=0}^{\infty}B_{n}P_{n},\] (A38)  
Substituting them back to Equations (A35) and (A36), we can have:  
\[\sum_{n=0}^{\infty}\{A_{n}[\lambda_{n}+(1+v)+\varphi]+B_{n}[\alpha\lambda_{n} +(1+v)+\varphi]\}P_{n}=0,\] (A39)  
\[\sum_{n=0}^{\infty}\biggl{\{} \begin{array}{c}A_{n}\left[\lambda_{n}^{2}+(1+ v)(\lambda_{n}+2)+\varphi(\lambda_{n}+2)\right]\\ +B_{n}[\alpha\lambda_{n}^{2}+(3+v)\alpha\lambda_{n}+2(1+v)-\varphi(\lambda_{n} -2)]\end{array} \biggr{\}}P_{n}=0.\] (A40)The Legendre functions form a complete set of functions. Therefore, the two series cannot vanish identically unless each coefficient vanishes. From Equation (A29) we can get:  
\[B_{n}=-\frac{A_{n}[\lambda_{n}+(1+v)+\varphi]}{\alpha\lambda_{n}+(1+v)+\varphi},\] (A41)  
and eliminating \(B_{n}\) from the above set of equations, Equation (A41) can then be written as:  
\[A_{n}\big{[}\alpha\lambda_{n}^{2}+(1+v)(\lambda_{n}+2)+\big{]}\] \[-\frac{A_{n}[\lambda_{n}+(1+v)+\varphi]}{[\alpha\lambda_{n}+(1+v )+\varphi][\alpha\lambda_{n}^{2}+(3+v)\alpha\lambda_{n}+2(1+v)-\varphi(\lambda _{n}-2)]},\] (A42)  
\[A_{n}\Big{(}1-v^{2}\Big{)}\lambda_{n}+\alpha\lambda_{n}\Big{[}\lambda_{n}^{2}+ 2\lambda_{n}+(1+v)^{2}\Big{]}-\varphi\lambda_{n}[\lambda_{n}+(1+3v)]=0.\] (A43)  
Buckling of the shells becomes possible if these equations, for some value of \(n\), yield for \(A_{n}\) and \(B_{n}\) a solution different than zero, which means a trivial solution; in other words, having a zero determinant of the system of equations. Thus:  
\[\Big{(}1-v^{2}\Big{)}\lambda_{n}+\alpha\lambda_{n}\Big{[}\lambda_{n}^{2}+2 \lambda_{n}+(1+v)^{2}\Big{]}-\varphi\lambda_{n}[\lambda_{n}+(1+3v)]=0,\] (A44)  
a solution of which is \(\lambda_{n}=0\), which corresponds to a value of \(n\) equals to unity. Substituting this value of \(\lambda_{n}\) into Equation (A40), one obtains:  
\[A_{1}=-B_{1}.\] (A45)  
Now, for \(\lambda_{n}\
eq 0\), other than zero:  
\[\varphi=\frac{\big{(}1-v^{2}\big{)}+\alpha\Big{[}\lambda_{n}^{2}+2\lambda_{n}+ \big{(}1+v\big{)}^{2}\Big{]}}{\lambda_{n}+(1+3v)},\] (A46)  
which yields for its minimum, or for \(\frac{\mathrm{d}\varphi}{\mathrm{d}\lambda_{n}}\) after simplification:  
\[\lambda_{n}^{2}+2(1+3v)\lambda_{n}-\frac{1+v^{2}}{\alpha}=0,\] (A47)  
\[\lambda_{n}=-(1-3v)\lambda_{n}+\sqrt{\frac{1-v^{2}}{\alpha}},\] (A48)  
\[\varphi_{min}=2\sqrt{(1-v^{2})\alpha}-6v\alpha,\] (A49)  
\[A_{n}\big{[}\alpha\lambda_{n}^{2}+(1+v)(\lambda_{n}+2)+\big{]}\]0 (A50)  
and \(\varphi_{min}\) yields the first \(p_{\sigma}\) critical load [33]:  
\[A_{n}\big{[}\alpha\lambda_{n}^{2}+(1+v)(\lambda_{n}+2)+\big{]}\]1 (A51)  
\[A_{n}\big{[}\alpha\lambda_{n}^{2}+(1+v)(\lambda_{n}+2)+\big{]}\]2 (A52)  
Equation (A51) was derived with the assumption that the shell wall thickness is specified, whereas the critical value of external pressure \(Q\) is the unknown quantity.