Page:Fourier - Théorie analytique de la chaleur, 1822.djvu/247

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CHAPITRE III.

toutes ces relations entre les quantités qui répondent à l’un des cas et celles qui répondent au cas suivant, on aura

$({\boldsymbol {c}})$

${\begin{array}{llllll}a_{1}=a_{2}(2^{2}-1)&&&&&\\a_{2}=a_{3}(3^{2}-1)&b_{2}=b_{3}(3^{2}-2^{2})&&&&\\a_{3}=a_{4}(4^{2}-1)&b_{3}=b_{4}(4^{2}-2^{2})&c_{3}=c_{4}(4^{2}-3^{2})&&&\\a_{4}=a_{5}(5^{2}-1)&b_{4}=b_{5}(5^{2}-2^{2})&c_{4}=c_{5}(5^{2}-3^{2})&d_{4}=d_{5}(5^{2}-4^{2})&&\\a_{5}=a_{6}(6^{2}-1)&b_{5}=b_{6}(6^{2}-2^{2})&c_{5}=c_{6}(6^{2}-3^{2})&d_{5}=d_{6}(6^{2}-4^{2})&e_{5}=e_{6}(6^{2}-5^{2})&\\a_{6}=a_{7}(7^{2}-1)&b_{6}=b_{7}(7^{2}-2^{2})&c_{6}=c_{7}(7^{2}-3^{2})&d_{6}=d_{7}(7^{2}-4^{2})&e_{6}=e_{7}(7^{2}-5^{2})&f_{6}=f_{7}(7^{2}-6^{2})\;;\\&&&&\mathrm {etc.} &&\\\end{array}}$

on aura aussi

${\begin{array}{llll}\mathrm {A} _{1}=2\mathrm {A} _{2}-\mathrm {B} _{2}&&&\qquad ({\boldsymbol {d}})\\\mathrm {A} _{2}=3\mathrm {A} _{3}-\mathrm {B} _{3}&\mathrm {B} _{2}=3\mathrm {B} _{3}-\mathrm {C} _{3}&&\\\mathrm {A} _{3}=4\mathrm {A} _{4}-\mathrm {B} _{4}&\mathrm {B} _{3}=4\mathrm {B} _{4}-\mathrm {C} _{4}&\mathrm {C} _{3}=4\mathrm {C} _{4}-\mathrm {D} _{4}&\\\mathrm {A} _{4}=5\mathrm {A} _{5}-\mathrm {B} _{5}&\mathrm {B} _{4}=5\mathrm {B} _{5}-\mathrm {C} _{5}&\mathrm {C} _{4}=5\mathrm {C} _{5}-\mathrm {D} _{5}&\mathrm {D} _{4}=5\mathrm {D} _{5}-\mathrm {E} _{5},\\&&\mathrm {etc.} \\\end{array}}$

On conclut des équations $(c)$ qu’en représentant par $a,\,b,\,c,\,d,\,e,\dots$ etc., les inconnues dont le nombre est infini, on doit avoir

${\begin{aligned}a&={\frac {a_{1}}{(2^{2}-1)(3^{2}-1)(4^{2}-1)(5^{2}-1)\cdots }}&\quad \\b&={\frac {b_{2}}{(3^{2}-2^{2})(4^{2}-2^{2})(5^{2}-2^{2})(6^{2}-2^{2})\cdots }}&\quad \\c&={\frac {c_{3}}{(4^{2}-3^{2})(5^{2}-3^{2})(6^{2}-3^{2})(7^{2}-3^{2})\cdots }}&\quad \\d&={\frac {d_{4}}{(5^{2}-4^{2})(6^{2}-4^{2})(7^{2}-4^{2})(8^{2}-4^{2})\cdots }}&\quad \\&\qquad \qquad \mathrm {etc.} &({\boldsymbol {e}})\\\end{aligned}}$