on aura
![{\displaystyle \operatorname {S} \alpha \gamma d\mu d\varpi \left[{\frac {\partial ^{2}u}{\partial t^{2}}}\sin(\varphi +\varpi )+2n\mu ^{2}{\frac {\partial u}{\partial t}}\cos(\varphi +\varpi )\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7c74b1640e7c2ef47af2cb77b84ddfa5b8e90f)
![{\displaystyle \left.+{\frac {\partial ^{2}v}{\partial t^{2}}}\mu {\sqrt {1-\mu ^{2}}}\cos(\varphi +\varpi )-2n{\frac {\partial v}{\partial t}}\mu {\sqrt {1-\mu ^{2}}}\sin(\varphi -\varpi )\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3124c90cdd75bde7bed2a64168e53c2750d26c9)
![{\displaystyle {\begin{aligned}=\quad \ &\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ \sin(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \mu }}\ -{\frac {\partial \mathrm {U} }{\partial \mu }}-{\frac {\partial f}{\partial \mu }}\right)\\-&\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\cos(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial \mathrm {U} }{\partial \varpi }}-{\frac {\partial f}{\partial \varpi }}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c37dfe34f88a9232149200135f114ad367914c5)
Si l’on substitue pour
l’ensemble des termes relatifs à l’angle
et si l’on observe qu’ici nous ne considérons que le cas dans lequel diffère très peu de
le premier membre de cette équation deviendra
![{\displaystyle -\alpha n^{2}\pi \operatorname {S} d\mu \cos(it+\varepsilon -\varphi )\left[\left(1-2\mu ^{2}\right)\mathrm {M} '-\mu {\sqrt {1-\mu ^{2}}}\mathrm {O} '\right]\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4598a23a87faabd2d68ed890571d410558abe70)
on aura donc, en n’ayant égard qu’aux termes dans lesquels
est à très peu près égal à ![{\displaystyle n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb)
![{\displaystyle {\begin{aligned}\operatorname {S} \alpha n^{2}&yd\mu d\varpi \mu {\sqrt {1-\mu ^{2}}}\sin(\varphi +\varpi )\\=&\quad \ \operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\sin(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \mu }}\ \ -{\frac {\partial \mathrm {U} }{\partial \mu }}-{\frac {\partial f}{\partial \mu }}\right)\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\cos(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial \mathrm {U} }{\partial \varpi }}-{\frac {\partial f}{\partial \varpi }}\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72197b7a1a37a0a02c4de637a8aff02ea508bcf8)
d’où l’on tire
![{\displaystyle {\begin{aligned}{\frac {d\mathrm {N} ''}{dt}}\sin \varphi -{\frac {d\mathrm {N} '}{dt}}\cos \varphi =&\quad \operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}{\frac {\partial f}{\partial \varpi }}\cos(\varphi +\varpi )\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ {\frac {\partial f}{\partial \mu }}\sin(\varphi +\varpi ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c783fe3a752779a9c10934f252c75ecb935201)
On trouvera de la même manière
![{\displaystyle {\begin{aligned}{\frac {d\mathrm {N} ''}{dt}}\cos \varphi +{\frac {d\mathrm {N} '}{dt}}\sin \varphi =&-\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}{\frac {\partial f}{\partial \varpi }}\sin(\varphi +\varpi )\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ {\frac {\partial f}{\partial \mu }}\cos(\varphi +\varpi ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70501602578e4327c496bd2627379e54d1585891)