259
RATIONNELLES.
donc, la suite qui vient du développement de la fraction
a pour terme général
![{\displaystyle {\begin{alignedat}{2}\theta (^{n)}&={\frac {n+1}{1}}.{\frac {n+2}{2}}.{\frac {n+3}{3}}&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots {\frac {n+m-1}{m-1}}.{\frac {\operatorname {Cos} .\left[(m+n)\phi -m.90^{\circ }\right]}{2^{m-1}\operatorname {Sin} .^{m}\phi }}&\\\\&-{\frac {n+1}{1}}.{\frac {n+2}{2}}.{\frac {n+3}{3}}&\ldots \ldots \ldots \ldots \ldots {\frac {n+m-2}{m-2}}.{\frac {m}{1}}.{\frac {\operatorname {Cos} .\left[(m+n-1)\phi -(m+1).90^{\circ }\right]}{2^{m}\operatorname {Sin} .^{m+1}\phi }}&\\\\&+{\frac {n+1}{1}}.{\frac {n+2}{2}}.{\frac {n+3}{3}}&\ldots \ldots \ldots {\frac {n+m-3}{m-3}}.{\frac {m}{1}}.{\frac {m+1}{2}}.{\frac {\operatorname {Cos} .\left[(m+n-2)\phi -(m+2).90^{\circ }\right]}{2^{m+1}\operatorname {Sin} .^{m+2}\phi }}&\\\\&-{\frac {n+1}{1}}.{\frac {n+2}{2}}.{\frac {n+3}{3}}&\ldots {\frac {n+m-4}{m-4}}.{\frac {m}{1}}.{\frac {m+1}{2}}.{\frac {m+2}{3}}.{\frac {\operatorname {Cos} .\left[(m+n-3)\phi -(m+3).90^{\circ }\right]}{2^{m+2}\operatorname {Sin} .^{m+3}\phi }}&\\\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/532598fdad469b34569ed4b87ef4cf8dd8a13ea6)
en continuant ainsi jusqu’à ce qu’on ait
termes.
5. On peut simplifier cette formule, en réduisant des termes
an même dénominateur, alors elle devient, après plusieurs
réductions
![{\displaystyle {\begin{alignedat}{2}\theta ^{(n)}&={\frac {n+m+1}{1}}.{\frac {n+m+2}{2}}.{\frac {n+m+3}{3}}.{\frac {n+m+4}{4}}\ldots &{\frac {n-2m-1}{m-1}}.{\frac {\operatorname {Sin} .(n+1)\phi }{2^{2m-2}\operatorname {Sin} .^{2m-1}\phi }}&\\\\&-{\frac {m-1}{1}}.{\frac {n+m+2}{2}}.{\frac {n+m+3}{3}}.{\frac {n+m+4}{4}}\ldots \ldots &{\frac {n-2m-1}{m-1}}.{\frac {\operatorname {Sin} .(n+3)\phi }{2^{2m-2}\operatorname {Sin} .^{2m-1}\phi }}&\\\\&+{\frac {m-1}{1}}.{\frac {m-2}{2}}.{\frac {n+m+3}{3}}.{\frac {n+m+4}{4}}\ldots \ldots \ldots &{\frac {n-2m-1}{m-1}}.{\frac {\operatorname {Sin} .(n+5)\phi }{2^{2m-2}\operatorname {Sin} .^{2m-1}\phi }}&\\\\&-{\frac {m-1}{1}}.{\frac {m-2}{2}}.{\frac {m-3}{3}}.{\frac {n+m+4}{4}}\ldots \ldots \ldots \ldots &{\frac {n-2m-1}{m-1}}.{\frac {\operatorname {Sin} .(n+7)\phi }{2^{2m-2}\operatorname {Sin} .^{2m-1}\phi }}&\\\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44c697db0abc0378dff5469a6784b82e78ed4c7)