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

 476 the present case gives, when $$2l$$ designates the length of the circuit, and the origin of the abscissæ is placed in its centre, 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, and then integrate, we obtain, since between the indicated limits and for the determination of $$v$$ the equation and, lastly, since $$u = u' + v$$ which equation, for $$\beta = 0$$, i. e. when it is not intended to take into consideration the influence of the atmosphere, passes into 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,