Pour cela je suppose
![{\displaystyle {\frac {m+nt^{2}}{1-m+(1-n)t^{2}}}=g^{2}{\frac {\mu +\nu \theta ^{2}}{1-\mu +(1-\nu )\theta ^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c203c98d1f3216b463fa5abba580de169c751d5a)
et
étant des coefficients indéterminés et
une nouvelle variable ; et je tire de là
![{\displaystyle t^{2}={\frac {g^{2}(1-m)\mu -m(1-\mu )+\left[g^{2}(1-m)\nu -m(1-\nu )\right]\theta ^{2}}{n(1-\mu )-g^{2}(1-n)\mu +\left[n(1-\nu )-g^{2}(1-n)\nu \right]\theta ^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08a104bb63d602ab7e5e584ad188d6460fc10906)
je suppose maintenant
![{\displaystyle g^{2}(1-m)\mu -m(1-\mu )=0,\quad n(1-\nu )-g^{2}(1-n)\nu =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c9b4036825a07d1a360f5bfad4d87d46e48325)
ce qui me donne
![{\displaystyle \mu ={\frac {m}{m+(1-m)g^{2}}},\quad \nu ={\frac {n}{n+(1-n)g^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9840f5c4fc193efea42dc3bb17084be798ede091)
j’aurai ainsi
![{\displaystyle t^{2}={\frac {g^{2}(1-m)\nu -m(1-\nu )}{n(1-\mu )-g^{2}(1-n)\mu }}\theta ^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7357923c954cd62166b8dac2a879e15299f586)
savoir, en substituant les valeurs précédentes de
et ![{\displaystyle \nu ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58da52de5fc23d5cc67b6a94171abb4420956f2)
![{\displaystyle t^{2}={\frac {g^{2}(1-m)+m}{g^{2}(1-n)+n}}\theta ^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbf499aa66f32fbd222b8a18a57d12f7f2050ec)
et de là
![{\displaystyle t=\theta {\sqrt {\frac {g^{2}(1-m)+m}{g^{2}(1-n)+n}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839c65ea6bc02c1511ff21cbcc6cb337a44a89cf)
de plus, à cause de
on aura
![{\displaystyle 1-m+(1-n)t^{2}={\frac {(1-m)n\mu +(1-n)m\nu \theta ^{2}}{n\mu }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ae1a6432f0476c4f90d8f172552e547a5038de)
mais les deux équations ci-dessus donnent
![{\displaystyle (1-m)\mu ={\frac {m(1-\mu )}{g^{2}}},\quad (1-n)\nu ={\frac {n(1-\nu )}{g^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5834adeba753d0638f2317d4cc5ca8c199dd3664)