# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 3 Calculation Example--Electric Conductor with Double Insulation
\begin{array}{l}p(n)=\frac{(\lambda_{1}-\lambda_{2})(\lambda_{3}-\lambda_{2} )}{4\lambda_{2}\lambda_{3}}\sinh 2(n\eta_{1})\sinh 2(n\eta_{2})+\\ +\Big{[}1+\frac{\lambda_{3}-\lambda_{2}}{\lambda_{3}}\sinh^{2}(n\eta_{2}) \Big{]}\Big{[}1+\frac{\lambda_{2}-\lambda_{1}}{\lambda_{2}}\sinh^{2}(n\eta_{1} )\Big{]},\\ q(n)=\frac{\lambda_{2}-\lambda_{3}}{2\lambda_{3}}\sinh 2(n\eta_{2})\Big{[}1+\frac{ \lambda_{2}-\lambda_{1}}{\lambda_{2}}\sinh^{2}(n\eta_{1})\Big{]}+\\ +\frac{\lambda_{1}-\lambda_{2}}{2\lambda_{3}}\sinh 2(n\eta_{1})\Big{[}1+\frac{ \lambda_{2}-\lambda_{3}}{\lambda_{2}}\sinh^{2}(n\eta_{2})\Big{]}.\end{array}  \tag{19}\]  
\[ \begin{array}{l}q(n)=\frac{\lambda_{2}-\lambda_{3}}{2\lambda_{3}}\sinh 2(n\eta_{2}) \Big{[}1+\frac{\lambda_{2}-\lambda_{1}}{\lambda_{2}}\sinh^{2}(n\eta_{1})\Big{]} +\\ +\frac{\lambda_{1}-\lambda_{2}}{2\lambda_{3}}\sinh 2(n\eta_{1})\Big{[}1+\frac{ \lambda_{2}-\lambda_{3}}{\lambda_{2}}\sinh^{2}(n\eta_{2})\Big{]}.\end{array}  \tag{20}\]  
Next, the coefficients \(A_{0}\), \(A_{n}\) in (8), (15) and (16) were determined. For this purpose, [PERSON]'s boundary condition (3) was used for \(N=3\). The summation of series (8), (15) and (16) was limited to a finite number \(L\) of terms. Then, the truncated series (16) was substituted in (3) for \(N=3\). Ultimately, the following was obtained:  
\[ \begin{array}{l}\frac{1}{c\sqrt{\cosh^{2}(\eta_{3})-\cos^{2}(\psi)}}\Big{(} -\frac{\delta_{1}c^{2}}{4\lambda_{3}}\sinh(2\eta_{1})+\frac{\delta_{1}c^{2}}{ 4\lambda_{1}}\cosh[2(\eta_{2}-\eta_{1})]\sinh[2(\eta_{2}-\eta_{3})]\cos(2\psi)- \\ -\frac{\delta_{1}c^{2}\lambda_{2}}{4\lambda_{1}\lambda_{3}}\sinh[2(\eta_{2}- \eta_{1})]\cosh[2(\eta_{2}-\eta_{3})]\cos(2\psi)+\\ +\sum\limits_{n=1}^{L}nA_{n}\{f(n)\sinh(n\eta_{3})+g(n)\cosh(n\eta_{3})\}\cos( n\psi)\Big{)}=\\ =-\frac{\delta_{3}}{\lambda_{3}}\Big{(}A_{0}-\frac{\delta_{1}c^{2}}{8\lambda_{ 1}}\cosh(2\eta_{1})+(\eta_{1}-\eta_{2})\frac{\delta_{1}c^{2}}{4\lambda_{2}} \sinh(2\eta_{1})+(\eta_{2}-\eta_{3})\frac{\delta_{1}c^{2}}{4\lambda_{3}}\sinh (2\eta_{1})-\\ -\frac{\delta_{1}c^{2}}{8\lambda_{1}}\cosh[2(\eta_{2}-\eta_{1})]\cosh[2(\eta_ {2}-\eta_{3})]\cos(2\psi)+\\ +\frac{\delta_{1}c^{2}\lambda_{2}}{8\lambda_{1}\lambda_{3}}\sinh[2(\eta_{2}- \eta_{1})]\sinh[2(\eta_{2}-\eta_{3})]\cos(2\psi)+\\ +\sum\limits_{n=1}^{L}A_{n}[p(n)\cosh(n\eta_{3})+q(n)\sinh(n\eta_{3})]\cos(n \psi).\end{array}  \tag{21}\]  
Next, relationship (21) was multiplied by \(\{\cos(m\psi)\}\) and then integrated by sides with respect to the angular coordinate \(\psi\) in the range \(<\)0, 2\(\times\). Thus, Equation (22) was obtained for \(m=1\), 2... \(L\). A subsequent Equation (23) was obtained by a double-sided integrating relationship (21) with respect to the angular coordinate \(\psi\) in the range \(<\)0, 2\(\times\). In this way, L + 1 equations were obtained with respect to the coefficients \(A_{0}\)... \(A_{n}\). \[ \begin{array}{l}\sum\limits_{n=1}^{L}A_{n}I_{1}(m,n)=I_{2}(m)\quad\text{for} \quad m=1,2\ldots\text{L},\\ \sum\limits_{n=1}^{L}A_{n}I_{3}(n)+A_{0}\frac{2\pi\alpha}{\lambda_{3}}=I_{4} \end{array}  \tag{22}\]  
where  
\[I_{1}(m,n)=r(n)\cdot h(m,n)\quad\text{for}\quad m\
eq n, \tag{24}\]  
\[I_{1}(m=n)=r(m)\cdot h(m=n)+\frac{\pi\alpha}{\lambda_{3}}[p(m)\cosh(m\eta_{3}) +q(m)\sinh(m\eta_{3})], \tag{25}\]  
\[ \begin{array}{l}I_{2}(m)=\frac{\delta_{1}c\sinh(2\eta_{1})}{4\lambda_{3}} \cdot h(m,n=0)-\frac{\delta_{1}c\cosh[2(\eta_{2}-\eta_{1})]\sinh[2(\eta_{2}- \eta_{3})]}{4\lambda_{1}}\cdot h(m,n=2)+\\ +\frac{\delta_{1}c\lambda_{2}\sinh[2(\eta_{2}-\eta_{1})]\cosh[2(\eta_{2}-\eta_ {3})]}{4\lambda_{1}\lambda_{3}}\cdot h(m,n=2)\qquad\text{for}\ m\
eq 2,\end{array}  \tag{26}\]\[