que
et
sont des quantités de l’ordre ![{\displaystyle \alpha ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f)
![{\displaystyle {\frac {d^{2}x_{\text{ı}}}{ds^{2}}}=\alpha {\frac {d^{2}u'_{\text{ı}}}{ds^{2}}}\sin \theta _{\text{ı}}+r_{\text{ı}}{\frac {d^{2}\theta _{\text{ı}}}{ds^{2}}}\cos \theta _{\text{ı}}-r_{\text{ı}}\sin \theta _{\text{ı}}{\frac {d^{2}\varphi _{\text{ı}}}{ds^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b6e4d438f818dacd28115d5efd37519fb9602a)
On a ensuite
![{\displaystyle \alpha {\frac {d^{2}u'_{\text{ı}}}{ds^{2}}}=\alpha {\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}{\frac {d\varphi _{\text{ı}}^{2}}{ds^{2}}}-\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}{\frac {d^{2}\theta _{\text{ı}}}{ds^{2}}}={\frac {\alpha {\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}}{\cos ^{2}\psi _{\text{ı}}}}-\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} \psi _{\text{ı}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54d90291a899555945f560b6e32b5cfa059af5bd)
de plus,
on aura donc, en substituant pour
et
leurs valeurs précédentes,
![{\displaystyle {\begin{aligned}{\frac {d^{2}x_{\text{ı}}}{ds^{2}}}=&(1-\alpha u'_{\text{ı}}){\frac {\sin ^{2}\psi _{\text{ı}}}{\cos \psi _{\text{ı}}}}+\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} ^{2}\psi _{\text{ı}}\sin \psi _{\text{ı}}\\\\&-{\frac {1}{\cos \psi _{\text{ı}}}}\left(1-\alpha u'_{\text{ı}}+\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} \psi _{\text{ı}}\right)+{\frac {\alpha {\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}}{\cos \psi _{\text{ı}}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e80f6b27b2c6cef005337701fa27b46d070893fd)
On a, comme on vient de le voir, en négligeant les puissances supérieures de ![{\displaystyle s,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5748cdb81bf00075de8e7e6828c343687513830)
![{\displaystyle {\rm {V-V_{\text{ı}}}}={\frac {s}{\cos \psi _{\text{ı}}}}\left(1-\alpha u'_{\text{ı}}+\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} \psi _{\text{ı}}-{\frac {\alpha {\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}}{\cos ^{2}\psi _{\text{ı}}}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fc26d503ee20ce8bb2dfbcb5a63e6b62da2423)
et
![{\displaystyle {\frac {dy}{ds}}=1\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe7bdad98dfe67685ba91d9f602013c151181b7)
on a donc
![{\displaystyle {\frac {dx'}{ds}}=s(1-\alpha u'_{\text{ı}}){\frac {\sin ^{2}\psi _{\text{ı}}}{\cos \psi _{\text{ı}}}}+\alpha s{\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} ^{2}\psi _{\text{ı}}\sin \psi _{\text{ı}}-\alpha s{\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}{\frac {\sin ^{2}\psi _{\text{ı}}}{\cos ^{3}\psi _{\text{ı}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13cfea41aebfeee7572ee22d3604e9df0bf52202)
on trouvera semblablement
![{\displaystyle {\frac {dz}{ds}}=s(1-\alpha u'_{\text{ı}})\sin \psi _{\text{ı}}-\alpha s{\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} ^{2}\psi _{\text{ı}}\cos \psi _{\text{ı}}+\alpha s{\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}{\frac {\sin \psi _{\text{ı}}}{\cos ^{2}\psi _{\text{ı}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0eae44a6de6a62d2c5e122e2626d0d56f93865b)
le cosinus de l’angle azimutal à l’extrémité de l’arc
sera ainsi
![{\displaystyle s\operatorname {tang} \psi _{\text{ı}}\left(1-\alpha u'_{\text{ı}}+\alpha {\frac {\partial u'_{\text{ı}}}{\partial \psi }}\operatorname {tang} \psi _{\text{ı}}-{\frac {\alpha {\frac {\partial ^{2}u'_{\text{ı}}}{\partial \varphi ^{2}}}}{\cos ^{2}\psi _{\text{ı}}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de06c03a81df6dc66980676c525b79633decc1c6)
Ce cosinus étant fort petit, il peut être pris pour le complément de