## 3 Electrode Station Configuration
### 10 th Electrode Station Configuration--Straight Frames Adapted Perimetrically to a Central Base
\[sin(\angle BON)=sin\bigg{(}\frac{\theta_{2}}{2}\bigg{)}=\frac{(BN)}{(OB)}=\frac{ \ell_{b2}/2}{R_{K2}}\Rightarrow\theta_{2}=2\times asin\bigg{(}\frac{\ell_{b2}}{2 \times R_{K2}}\bigg{)}\simeq 2\times asin\bigg{(}\frac{ds_{c}}{2\times R_{K2}} \bigg{)} \tag{49}\]  
From the combination of Equations (47)-(49), the following equation is determined, given the frame length \(\ell_{f}\), the minimum distance between consecutive frames \(ds_{c}\), and the arc angle between consecutive frames \(\theta_{f}\) (from relation (39)), which can be solved through numerical analysis, with respect to the radius \(R_{K2}\):  
\[\theta_{f}=2\times asin\bigg{(}\frac{\ell_{f}}{2\times R_{K2}}\bigg{)}+2\times asin \bigg{(}\frac{ds_{c}}{2\times R_{K2}}\bigg{)} \tag{50}\]  
Alternatively, the necessary \"gross\" radius \(R_{K2}\) of the central base (so as not to apply a numerical analysis) is calculated as follows: Initially, the angle \(\theta_{3}\) is calculated through the angles of the quadrilateral OMPR as supplementary to \(\theta_{f}\):  
Figure 20:0  
Applying the law of cosines for the angle \(\theta_{3}\) of the BPC triangle, the length \(\ell_{1}\) is calculated:  
\[ \begin{array}{c}(BC)^{2}=(BP)^{2}+(CP)^{2}-2\times(BP)\times(CP)\times cos( \angle BPC)\Rightarrow\ell_{b1}{}^{2}=\ell_{1}{}^{2}+\ell_{1}{}^{2}-2\times \ell_{1}\times\ell_{1}\times cos(\theta_{3})\Rightarrow\\ \ell_{b1}{}^{2}=2\times\ell_{1}{}^{2}-2\times\ell_{1}{}^{2}\times cos\Big{(}1 80^{\circ}-\theta_{f}\Big{)}\Rightarrow\ell_{1}=\frac{\ell_{b1}}{\sqrt{2 \times\big{(}1+cos(\theta_{f})\big{)}}}\simeq\frac{ds_{c}}{\sqrt{2\times\big{(} 1+cos(\theta_{f})\big{)}}}\end{array}  \tag{52}\]  
Applying the law of cosines for the angle \(\theta_{3}\) of the APC triangle, the length \(\ell_{2}\) is calculated:  
\[ \begin{array}{c}(AC)^{2}=(AP)^{2}+(CP)^{2}-2\times(AP)\times(CP)\times cos( \angle APC)\Rightarrow\ell_{2}{}^{2}=(\ell_{b1}+\ell_{1})^{2}+\ell_{1}{}^{2}-2 \times(\ell_{b1}+\ell_{1})\times\ell_{1}\times cos(\theta_{3})\\ \Rightarrow\ell_{2}{}^{2}=(\ell_{b1}+\ell_{1})^{2}+\ell_{1}{}^{2}-2\times( \ell_{b1}+\ell_{1})\times\ell_{1}\times cos\Big{(}180^{\circ}-\theta_{f}\Big{)} \Rightarrow\\ \ell_{2}=\sqrt{(\ell_{b1}+\ell_{1})^{2}+\ell_{1}{}^{2}+2\times(\ell_{b1}+ \ell_{1})\times\ell_{1}\times cos\Big{(}\theta_{f}\Big{)}}\simeq\sqrt{\Big{(} \ell_{f}+\ell_{1}\Big{)}^{2}+\ell_{1}{}^{2}+2\times\Big{(}\ell_{f}+\ell_{1} \Big{)}\times\ell_{1}\times cos\Big{(}\theta_{f}\Big{)}}\end{array}  \tag{53}\]  
Through the definition of sine in OAC triangles, the \"gross\" radius \(R_{K2}\) of the central base is obtained:  
\[sin(\angle AOQ)=sin\bigg{(}\frac{\theta_{f}}{2}\bigg{)}=\frac{(QQ)}{(OA)}= \frac{\ell_{2}/2}{R_{K2}}\Rightarrow R_{K2}=\frac{\ell_{2}}{2\times sin\Big{(} \theta_{f}/2\Big{)}} \tag{54}\]  
So, through the successive application of Equations (52)-(54), the calculation of \(R_{K2}\) is carried out directly. Then, by choosing an appropriate value of the gross radius \(R_{K2}\)/ of the central base, through the definition of sine in the OBM triangle, the final angle \(\theta_{1}\)/ is obtained:  
\[sin(\angle MOB)=sin\bigg{(}\frac{\theta_{1}{}^{\prime}}{2}\bigg{)}=\frac{(BM)} {(OB)}=\frac{\ell_{b1}/2}{R_{K2}}\Rightarrow\theta_{1}=2\times asin\bigg{(} \frac{\ell_{b1}}{2\times R_{K2}}\bigg{)}\simeq 2\times asin\bigg{(}\frac{\ell_{f}}{2 \times R_{K2}}\bigg{)} \tag{55}\]  
Therefore, the distance of any frame \(d_{Off}\), from the centre O of the arrangement (with the help of the definition of cosine on the final angle \(\theta_{1}{}^{\prime}/2\)), is equal to:  
\[cos(\angle MOB)=cos\bigg{(}\frac{\theta_{1}{}^{\prime}}{2}\bigg{)}=\frac{(OM)} {(OB)}=\frac{d_{Off}}{R_{K2}{}^{\prime}}\Rightarrow d_{Off}=R_{K2}{}^{\prime} \times cos\bigg{(}\frac{\theta_{1}{}^{\prime}}{2}\bigg{)} \tag{56}\]  
The respective surface \(S_{Gr-X}\) is calculated through Equation (40). Essentially, it is a variation in the central base solution, with the placement of the straight frames around the perimeter, instead of radially.