## 6 Conclusions
Then they can be written as:  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]2  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]3  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]4  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]5  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]6  
And then we get:  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]7  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]8  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\]9  
\[\frac{dM_{x}}{d\theta}+\big{(}M_{x}-M_{y}\big{)}\cot\theta-Q_{x}R+M_{y}\bigg{(} \frac{u}{R}+\frac{dw}{Rd\theta}\bigg{)}=0.\]0  
Now, introducing two dimensionless parameters, \(\alpha\) and \(\varphi\)--which are defined as \(\alpha=\frac{D\big{(}1-v^{2}\big{)}}{R^{2}Eh_{0}}=\frac{h_{0}}{12R^{2}}\) and \(\varphi=\frac{pR\big{(}1-v^{2}\big{)}}{2 Eh_{0}}\)--and using the elastic law to express the forces and moments in terms of \(u\) and \(w\), one obtains the differential equations of equilibrium ((A7) and (A8)):  
\[ \begin{array}{c}(1+\alpha)\Big{[}\frac{d^{2}u}{d\theta^{2}}+\cot\theta\frac {du}{d\theta}-\big{(}v+\cot^{2}\theta\big{)}u\Big{]}-(1+v)\frac{dw}{d\theta}+ \alpha\Big{[}\frac{d^{2}w}{d\theta^{3}}+\cot\theta\frac{d^{2}w}{d\theta^{2}} -\big{(}v+\cot^{2}\theta\big{)}\frac{dw}{d\theta}\Big{]}\\ \qquad\qquad\qquad\qquad\pm\varphi\Big{(}u-\frac{dw}{d\theta}\Big{)}=0,\end{array}  \tag{28}\]  
\[ \begin{array}{c}(1+v)\Big{[}\frac{du}{d\theta}+u\cot\theta-2w\Big{]}+\\ +\alpha\Big{[}\frac{d^{2}u}{d\theta^{3}}-2\cot\theta\frac{d^{2}u}{d\theta^{2} }+\big{(}1+v+\cot^{2}\theta\big{)}\Big{(}\frac{du}{d\theta}+\frac{d^{2}w}{d \theta^{2}}\Big{)}-\cot\theta\big{(}2-v+\cot^{2}\theta\big{)}(u+1)-\frac{d^{ 4}w}{d\theta^{4}}\\ \qquad\qquad\qquad\qquad\qquad+-2\cot\theta\frac{d^{2}w}{d\theta^{3}}- \varphi\Big{[}-uccot\theta-\frac{dw}{d\theta}+4w+cot\frac{dw}{d\theta}+\frac{d ^{2}w}{d\theta^{2}}\Big{]}=0.\end{array}  \tag{29}\]These two equations may be simplified by neglecting, in comparison with unity, the first term, since the shell is thin, and therefore, the \(\frac{h_{0}}{R}\) ratio is very small. Moreover, due largely to angular displacement \(\chi\), we make good use of this situation by introducing an auxiliary variable \(u\), such that \(u=-\frac{d\varphi}{d\theta}\). Thus, the expressions in the brackets in Equation (A29) become identical. Then, using the mathematical operator \(H\), it turns to:  
\[H(\ \ )=\frac{d^{2}(\ldots)}{d\theta^{2}}+\cot\theta\frac{d(\ldots)}{d\theta}+2 (\ldots),\] (A30)  
\[\frac{d}{d\theta}[H(\psi)+\alpha H(w)-(1+v)(\psi+w)-\alpha(1+v)w-\varphi(\psi +w)]=0.\] (A31)  
The fourth term, containing the factor, may be neglected in comparison with the third in Equation (A31). Integrating the Equation (A31) with respect to \(\theta\) and assuming that the constant of integration is equal to zero, we obtain:  
\[H(\psi)+\alpha H(w)-(1+v)(\psi+w)-\varphi(\psi+w)=0,\] (A32)  
\[