377
DES SUITES.
et par le Corollaire III, pour
impair,
![{\displaystyle {\frac {a\operatorname {Sin} .^{n}t}{1}}+{\frac {a^{2}\operatorname {Sin} .^{n}2t}{1.2}}+{\frac {a^{3}\operatorname {Sin} .^{n}3t}{1.2.3}}+{\frac {a^{4}\operatorname {Sin} .^{n}4t}{1.2.3.4}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc41b20a74ac4851e478c0197c7c3fcb7f80291c)
![{\displaystyle =\pm {\frac {1}{2^{n-1}}}\left\{e^{a\operatorname {Cos} .nt}.\operatorname {Sin} .[a\operatorname {Sin} .nt]-{\frac {n}{1}}e^{a\operatorname {Cos} .(n-2)t}.\operatorname {Sin} .\left[a\operatorname {Sin} .(n-2)t\right]\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1eb44455f41f6efa65adcf392f5476c59cc43c)
![{\displaystyle \left.+{\frac {n}{1}}.{\frac {n-1}{2}}e^{a\operatorname {Cos} .(n-4)t}.\operatorname {Sin} .\left[a\operatorname {Sin} .(n-4)t\right]+\ldots \right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fa2e41dc37aea86fc336ab4fbf3fa5b1966293)
14. Si, par exemple, on suppose
les première, deuxième, quatrième et cinquième formules deviendront, en ayant toujours égard aux limitations prescrites pour les seconds membres,
1.o
![{\displaystyle 1+{\frac {a\operatorname {Cos} .t\operatorname {Cos} .u}{1}}+{\frac {a^{2}\operatorname {Cos} .2t\operatorname {Cos} .2u}{1.2}}+{\frac {a^{3}\operatorname {Cos} .3t\operatorname {Cos} .3u}{1.2.3}}+{\frac {a^{4}\operatorname {Cos} .4t\operatorname {Cos} .4u}{1.2.3.4}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63a0ef90c3a9f6e3c2d50805d8d89f4a0661e93e)
![{\displaystyle =+{\tfrac {1}{2}}\left\{e^{a\operatorname {Cos} .(t+u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u)\right]+e^{a\operatorname {Cos} .(t-u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t-u)\right]\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa12a35ed3b434b6af12a9befc4d68f8c69015f)
2.o
![{\displaystyle {\frac {a\operatorname {Sin} .t\operatorname {Sin} .u}{1}}+{\frac {a^{2}\operatorname {Sin} .2t\operatorname {Sin} .2u}{1.2}}+{\frac {a^{3}\operatorname {Sin} .3t\operatorname {Sin} .3u}{1.2.3}}+{\frac {a^{4}\operatorname {Sin} .4t\operatorname {Sin} .4u}{1.2.3.4}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9930139726b0c37189afed5ea65ea1bb734edbd3)
![{\displaystyle =-{\tfrac {1}{2}}\left\{e^{a\operatorname {Cos} .(t+u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u)\right]-e^{a\operatorname {Cos} .(t-u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t-u)\right]\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cf805d05f8d25062f1c7a44da05234ffa32d4c)
3.o ![{\displaystyle 1+{\frac {a\operatorname {Cos} .^{2}t}{1}}+{\frac {a^{2}\operatorname {Cos} .^{2}2t}{1.2}}+{\frac {a^{3}\operatorname {Cos} .^{2}3t}{1.2.3}}+\ldots =+{\frac {1}{2}}\left\{e^{a\operatorname {Cos} .2t}.\operatorname {Cos} .(a\operatorname {Sin} .2t)+e^{a}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41ae093653fcf46b2cd23183c658c182b069485)
4.o ![{\displaystyle {\frac {a\operatorname {Sin} .^{2}t}{1}}+{\frac {a^{2}\operatorname {Sin} .^{2}2t}{1.2}}+{\frac {a^{3}\operatorname {Sin} .^{2}3t}{1.2.3}}+\ldots =-{\frac {1}{2}}\left\{e^{a\operatorname {Cos} .2t}.\operatorname {Cos} .(a\operatorname {Sin} .2t)-e^{a}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ae355655e23cc146ac3b3d5c0d93879c046078)
Ces deux dernières formules avaient déjà été données par M. Stein, à la page 112 du présent volume.
15. Si l’on suppose
les première, troisième, quatrième et sixième formules deviendront