Page:Scientific Memoirs, Vol. 2 (1841).djvu/487

then remains for the determination of the function $v$, which still possesses the same form as the equation (✳), but differs from it in this respect, that $v$ is a function of $x$, and $t$ of a different nature from $u$, by which its final determination is much facilitated.

The integral of the equation (☽), in the form in which it was first obtained by Laplace, is

$v={\frac {e^{-\chi '\beta ^{2}t}}{\sqrt {\pi }}}\int e^{-y^{2}}f(x+2y{\sqrt {\chi 't}})~dy$,

(☿)

where $e$ represents the base of the natural logarithms, $\pi$ the ratio of the circumference of a circle to its diameter, and $f$ an arbitrary function to be determined from the peculiar nature of each problem, while the limits of the integration must be taken from $y=-\infty$ to $y=+\infty$. For $t=0$ we have $v=fx$, because between the indicated limits $fe^{-y^{2}}~dy={\sqrt {\pi }}$, whence it results that if we know how to find the function $v$ in the case where $f=0$, we should thereby likewise discover $f~x$, consequently the arbitrary function $f$. Now in general $v=u-u'$; but if we reckon the time $t$ from the moment when, by the contact at the two extremities of the circuit, the tension originates, then $u$, when $t=0$ has evidently fixed values only at these extremities, at all other places of the circuit $u$ is $=0$; accordingly, in the whole extent of the circuit $v=-u'$ in general when $t=0$; only at the extremities of the circuit at the same time $v=u-u'$. If, therefore, we imagine a circuit left from the first moment of contact entirely to itself, then $v$ constantly $=0$ at its extremities, so that therefore in the interior of the circuit $v=-u'$, when $t=0$, and at its extremities $v=0$. Since, in accordance with our previous inquiries, $u'$ may be regarded as known for each place of the circuit, this likewise applies to $v$ when $t=0$; we know then the form of the arbitrary function $f~x$, so long as $x$ belongs to a point in the circuit.

However, the integral given for the determination of $v$ requires the knowledge of the function $f~x$ for all positive and negative values of $x$; we are thus compelled to give, by transformation, such as the researches respecting the diffusion of heat have made us acquainted with, such a form to the above equation that only pre-supposes the knowledge of the function $f~x$ for the extent of the circuit. The transformation applicable to