Page:Scientific Memoirs, Vol. 2 (1841).djvu/488

the present case gives, when $2l$ designates the length of the circuit, and the origin of the abscissæ is placed in its centre^[1],

${\begin{aligned}v=&{\frac {e^{-\chi '\beta ^{2}t}}{l}}\left[\Sigma \left(e^{\frac {-\chi 'i^{2}\pi ^{2}t}{l^{2}}}\cdot \sin {\frac {i\pi x}{l}}\int \sin {\frac {i\pi y}{l}}f\,y\,dy\right)\right.\\&+\left.\Sigma \left(e^{\frac {-(2i-1)^{2}\pi ^{2}t}{4l^{2}}}\cos {\frac {(2i-1)\pi x}{2l}}\int \cos {\frac {(2i-1)\pi y}{2l}}f\,y\,dy\right)\right]\end{aligned}}$

where the sums must be taken from $i=1$ to $i=\infty$, and the integrals from $y=-l$ to $y=+l$. If we now substitute in this equation for $f\,x$ its value $-u'$, whereby according to our supposition in the preceding paragraph, if a represents the tension at the place of contact,

$u'={\frac {{\frac {1}{2}}a(e^{\beta x}-e^{-\beta x})}{e^{\beta l}-e^{-\beta l}}},$

and then integrate, we obtain, since between the indicated limits

${\frac {1}{2}}a\int \sin {\frac {i\pi y}{l}}\cdot {\frac {e^{\beta y}-e^{-\beta y}}{e^{\beta l}-e^{-\beta l}}}\cdot dy=-{\frac {ai\pi l\cos i\pi }{i^{2}\pi ^{2}+\beta ^{2}l^{2}}},$

and

${\frac {1}{2}}a\int {\frac {e^{\beta y}-e^{-\beta y}}{e^{\beta l}-e^{-\beta l}}}\cdot \cos {\frac {(2i-1)\pi y}{2l}}\cdot dy=0,$

for the determination of $v$ the equation

$v=a\cdot e^{-\chi '\beta ^{2}t}\Sigma \left({\frac {i\pi \sin {\frac {i\pi (l+x)}{l}}}{i^{2}\pi ^{2}+\beta ^{2}l^{2}}}\cdot e^{\frac {-\chi '\pi ^{2}i^{2}t}{l^{2}}}\right),$

and, lastly, since $u=u'+v$

$u={\frac {{\frac {1}{2}}a(e^{\beta x}-e^{-\beta x})}{e^{\beta l}-e^{-\beta l}}}+a\cdot e^{-\chi '\beta ^{2}t}\times \Sigma \left({\frac {i\pi \sin {\frac {i\pi (l+x)}{l}}}{i^{2}\pi ^{2}+\beta ^{2}l^{2}}}\cdot e^{\frac {-\chi '\pi ^{2}i^{2}t}{l^{2}}}\right),$

which equation, for $\beta =0$, i. e. when it is not intended to take into consideration the influence of the atmosphere, passes into

$u={\frac {a}{2l}}x+a\Sigma \left({\frac {1}{i\pi }}\sin {\frac {i\pi (l+x)}{l}}\cdot e^{\frac {-\chi '\pi ^{2}i^{2}t}{l^{2}}}\right).$

It is easily perceived that the value of the second member to the right in the equations which have been found for the determination of $u$, becomes smaller and smaller as the time increases,

1. See Journal de l'Ecole Polytechnique, cap. xix. p. 53.