Page:Scientific Memoirs, Vol. 2 (1841).djvu/471

from one extremity to the other, to the extent of an entire tension.

b. When any place of the circuit is disposed by any circumstance to change its electric state, all the other places of the circuit change theirs at the same time, and to the same amount.

16. We will now imagine a galvanic circuit, composed of two parts, $P$ and $P'$, at whose two points of contact a different electric tension occurs, which case comprises in it the thermal circuit. If we call $u$ the electroscopic force of the part $P$, and $u'$ that of the part $P'$, then, according to the preceding paragraph, as here, the case there noticed is repeated twice, in consequence of the equation (c),

$u=fx+c$

for the part $P$, and

$u'=f'x+c'$

for the part $P'$, where $f$, $c$, $f'$, $c'$ are any constant magnitudes to be deduced from the peculiar circumstances of our problem, and each equation is only valid so long as the abscissæ refer to that part to which the equations belong. If we now place the origin of the abscissæ at one of the places of contact of the part $P$, and suppose the direction of the abscissæ in this part to proceed inwards; moreover, designate by $l$ the length of the part $P$, and by $l'$ that of $P'$; and, lastly, represent by $u'_{2}$ and $u_{1}$ the values of $u$ and $u'$ at the place of contact where $x=0$, and by $u_{2}$ and $u'_{1}$ the values of $u$ and $u'$ at the place of contact where $x=l$, we then obtain

${\begin{aligned}u'_{2}&=f'(l+l')+c'&u_{1}&=c\\u_{2}&=fl+c&u'_{1}&=f'l+c'.\end{aligned}}$

If we now designate by $a$ the tension which occurs at the place of contact where $x=0$, and by $a'$ that which occurs at the place of contact where $x=l$; and if we once for all assume, for the sake of uniformity, that the tension at each individual place of contact always expresses the value which is obtained when we deduct the electroscopic force of one extremity from the electroscopic force of that extremity belonging to the place in question, upon which the abscissa falls before the abrupt change takes place—(it is not difficult to perceive that this general rule contains that advanced in the preceding paragraph, and which, in fact, expresses nothing more than that the tensions of such