![{\displaystyle 4S=2U+2V=\operatorname {Log} .\left(1+2u\operatorname {Cos} .y+u^{2}\right)+\operatorname {Log} .\left(1+2v\operatorname {Cos} .y+v^{2}\right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d9c1bdcaa89fc20cfc430636949b6e781ebedb)
c’est-à-dire,
![{\displaystyle 4S=\operatorname {Log} .\left(1+2u\operatorname {Cos} .y+u^{2}\right)\left(1+2v\operatorname {Cos} .y+v^{2}\right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76888b8512510083be376da4132b784805b028e6)
en développant et se rappelant que
cela donnera
![{\displaystyle 4S=\operatorname {Log} .\left[2+u^{2}+v^{2}+4(u+v)\operatorname {Cos} .y+4\operatorname {Cos} .^{2}y\right]\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca93cf8094d76ad7db85a7b000deaddac481b159)
mais
![{\displaystyle u+v=2\operatorname {Cos} .x,\qquad u^{2}+v^{2}=(u+v)^{2}-2=4\operatorname {Cos} .^{2}x-2\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b7b11954be48ae990de16df46f77e5b7d3899a)
donc
![{\displaystyle 4S=\operatorname {Log} .4\left(\operatorname {Cos} .^{2}x+2\operatorname {Cos} .x\operatorname {Cos} .y+\operatorname {Cos} ^{2}y\right)=\operatorname {Log} .4(\operatorname {Cos} .x+\operatorname {Cos} .y)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930f5319381045880b1f9c0c9caa24dd6a67c021)
ou
![{\displaystyle 4S=2\operatorname {Log} .(2\operatorname {Cos} .x+2\operatorname {Cos} .y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/557fa9ef9640fe4a7bdc56750f1179ad53844318)
d’où finalement
![{\displaystyle S={\frac {1}{2}}\operatorname {Log} .2(\operatorname {Cos} .x+\operatorname {Cos} .y)={\frac {1}{2}}\operatorname {Log} .4\operatorname {Cos} .{\frac {1}{2}}(x+y)\operatorname {Cos} .{\frac {1}{2}}(x-y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53c2e76f08befe45d37037ede3e04e512db6ebd6)
Nous ayons donc, en résumé,
1.o ![{\displaystyle {\frac {a\operatorname {Cos} .x}{1}}-{\frac {a^{3}\operatorname {Cos} .3x}{3}}+{\frac {a^{5}\operatorname {Cos} .5x}{5}}-\ldots ={\frac {1}{2}}\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {2a\operatorname {Cos} .x}{1-a^{2}}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb22f37b1d8b0b820d2d9da9ce8d75174ffac2c4)
2.o ![{\displaystyle {\frac {\operatorname {Cos} .x}{1}}+{\frac {1}{2}}{\frac {\operatorname {Cos} .3x}{3}}+{\frac {1.3}{2.4}}{\frac {\operatorname {Cos} .5x}{5}}+\ldots ={\frac {1}{2}}\operatorname {Arc} .(\operatorname {Cos} .=2\operatorname {Sin} .x-1),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb804c24dc4e0883f21e830da08a85a7f903ee80)
3.o ![{\displaystyle {\frac {\operatorname {Cos} .x\operatorname {Cos} .y}{1}}-{\frac {\operatorname {Cos} .2x\operatorname {Cos} .2y}{2}}+{\frac {\operatorname {Cos} .3x\operatorname {Cos} .3y}{3}}-\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb9de4f94011bc8d54ccb8a66cb49052719d0c0)
![{\displaystyle ={\frac {1}{2}}\operatorname {Log} .4\operatorname {Cos} .{\frac {1}{2}}(x+y)\operatorname {Cos} .{\frac {1}{2}}(x-y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd22f1ba408882d719281e95c12b1f546981088)
Il est aisé de parvenir, en suivant la même marche, à des sommes de séries beaucoup plus compliquées. Nous nous bornerons à en rapporter deux exemples, en nous dispensant de développer les calculs qui sont en tout semblables à ceux qu’on a vu ci-dessus.