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

Rh 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 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