Page:Annales de mathématiques pures et appliquées, 1813-1814, Tome 4.djvu/186

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PROBLÈMES

rapports différentiels partiels ${\frac {\operatorname {d} ^{2}t}{2\operatorname {d} \lambda ^{2}}},{\frac {\operatorname {d} ^{2}t}{2\operatorname {d} \lambda \operatorname {d} \mu }},{\frac {\operatorname {d} ^{2}t}{2\operatorname {d} \mu ^{2}}}$. Faisant, pour abréger

$P=1-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi ,\qquad Q=1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi ,$

ce qui donne

$\mathrm {\operatorname {d} P} =-\operatorname {d} \lambda \operatorname {Cos} .\lambda \operatorname {Cos} .\phi +\operatorname {d} \phi \operatorname {Sin} .\lambda \operatorname {Sin} .\phi ,$

$\mathrm {\operatorname {d} Q} =-\operatorname {d} \mu \operatorname {Cos} .\psi \operatorname {Cos} .\phi +\operatorname {d} \psi \operatorname {Sin} .\mu \operatorname {Sin} .\psi ,$

on parviendra ainsi à donner une forme un peu plus abrégée aux deux rapports ${\frac {\operatorname {d} t}{\operatorname {d} \lambda }},{\frac {\operatorname {d} t}{\operatorname {d} \mu }},$ lesques deviendront

$2\varpi \left({\frac {\operatorname {d} t}{\operatorname {d} \lambda }}\right)=+{\frac {pq\operatorname {Cos} .^{2}\lambda \operatorname {Cos} .^{3}\mu \operatorname {Sin} .\phi (1+P)}{qP^{2}\operatorname {Cos} .^{3}\mu -pQ^{2}\operatorname {Cos} .^{3}\lambda }},$

$2\varpi \left({\frac {\operatorname {d} t}{\operatorname {d} \mu }}\right)=-{\frac {pq\operatorname {Cos} .^{2}\mu \operatorname {Cos} .^{3}\lambda \operatorname {Sin} .\psi (1+Q)}{qP^{2}\operatorname {Cos} .^{3}\mu -pQ^{2}\operatorname {Cos} .^{3}\lambda }}.$

Mais il est convenable d’abréger encore. Désignons par $F,M,N$ le dénominateur commun et les numérateurs de ces deux valeurs, de manière qu’on ait

$2\varpi \left({\frac {\operatorname {d} t}{\operatorname {d} \lambda }}\right)={\frac {M}{F}},\qquad 2\varpi \left({\frac {\operatorname {d} t}{\operatorname {d} \mu }}\right)=-{\frac {N}{F}}\,;$

les différentiations partielles nous apprendront que

$\left({\frac {\operatorname {d} F}{\operatorname {d} \lambda }}\right)=-2qP\operatorname {Cos} .^{3}\mu \operatorname {Cos} .\phi +3pQ^{2}\operatorname {Sin} .\lambda \operatorname {Cos} .^{2}\lambda$

$+\left({\frac {\operatorname {d} \phi }{\operatorname {d} \lambda }}\right)\left(2qP\operatorname {Cos} .^{3}\mu \operatorname {Sin} .\lambda \operatorname {Sin} .\phi -2pQ\operatorname {Cos} .^{3}\lambda \operatorname {Sin} .\mu \operatorname {Sin} .\psi \right)$,

$\left({\frac {\operatorname {d} F}{\operatorname {d} \mu }}\right)=+2pQ\operatorname {Cos} .\mu \operatorname {Cos} .^{3}\lambda \operatorname {Cos} .\psi -3qP^{2}\operatorname {Sin} .\mu \operatorname {Cos} .^{2}\mu$

$+\left({\frac {\operatorname {d} \psi }{\operatorname {d} \mu }}\right)\left(2qP\operatorname {Cos} .^{3}\mu \operatorname {Sin} .\lambda \operatorname {Sin} .\phi -2pQ\operatorname {Cos} .^{3}\lambda \operatorname {Sin} .\mu \operatorname {Sin} .\psi \right)$,

$\left({\frac {\operatorname {d} M}{\operatorname {d} \lambda }}\right)=-pq\operatorname {Cos} .\lambda \operatorname {Cos} .^{3}\mu \operatorname {Sin} .\phi \left(4\operatorname {Sin} .\lambda +\operatorname {Cos} ^{2}\lambda .\operatorname {Cos} .\phi -2\operatorname {Sin} .^{2}\lambda \operatorname {Cos} .\phi \right)$