![{\displaystyle \left.{\begin{aligned}&{\frac {\operatorname {d} z}{\operatorname {d} s}}{\frac {\operatorname {d} x}{\operatorname {d} s}}+(z+A){\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}-N\operatorname {Cos} .\alpha =0,\\\\&{\frac {\operatorname {d} z}{\operatorname {d} s}}{\frac {\operatorname {d} y}{\operatorname {d} s}}+(z+A){\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}-N\operatorname {Cos} .\beta =0,\\\\&{\frac {\operatorname {d} z}{\operatorname {d} s}}{\frac {\operatorname {d} z}{\operatorname {d} s}}+(z+A){\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}-N\operatorname {Cos} .\gamma =1.\end{aligned}}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba731dcfd98cb3f131069459f776f769739d1ca0)
(13)
Pour tirer facilement de ces équations la valeur de la pression normale
au point
, nous prendrons la somme de leurs produits respectifs par
ce qui donnera, en ayant égard aux relations (4) et (5),
![{\displaystyle (z+A)\left(P{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+Q{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}+R{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\right)-{\frac {N}{V}}=R\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e969f2778d8ac4eb3b10c1c97908961e4930a978)
d’où on tirera
![{\displaystyle N=V\left\{(z+A)\left(P{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+Q{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}+R{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\right)-R\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c274b5934f5474b8279a2ebb9e3396ff59c37f83)
ou encore
![{\displaystyle N={\frac {(z+A)\left(P{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+Q{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}+R{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\right)-R}{\sqrt {P^{2}+Q^{2}+R^{2}}}}.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3466e3738035161d32b482c0d5bc3de51fba778)
(14)
En substituant cette valeur, ainsi que la valeur (3) de
dans la dernière des équations (13), nous obtiendrons, pour l’équation différentielle du second ordre d’une surface qui doit couper la surface
suivant la chaînette cherchée,
![{\displaystyle R=(z+A)\left(P{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+Q{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/373d8b01602b2ebb1faaa068c47c8bcdfe450cb4)
![{\displaystyle =\left(P^{2}+Q^{2}\right)\left\{(z+A){\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}-1\right\}+\left(P^{2}+Q^{2}+R^{2}\right)\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}.\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2ed5ddb2ee284d7d10c07da7bb90dbf5627efd)
(15)