Page:1860 Fizeau en.djvu/10

same manner as before. For the difference of path we have the value

$\Delta =E\left\{{\frac {v}{v'-{\frac {v^{2}-v'^{2}}{v^{2}}}u}}-{\frac {v}{v'+{\frac {v^{2}-v'^{2}}{v^{2}}}u}}\right\}$,

which by reduction and transformation becomes

$\Delta =2E{\frac {u}{v}}\left\{{\frac {v^{2}-v'^{2}}{v'^{2}-u^{2}\left({\frac {v^{2}-v'^{2}}{v^{2}}}\right)^{2}}}\right\}$.

Taking into consideration the smallness of $u$ with respect to $v'\left({\tfrac {u}{v'}}={\tfrac {1}{33000000}}\right)$, and the circumstance that the coefficient of $u^{2}$ differs little from unity, the term in $u^{2}$ may, without appreciable error, be neglected, and the above expression considerably simplified. In fact, if $m$ be the index of refraction, and $L={\tfrac {1}{2}}E$ the length of each tube, we have approximately

$\Delta =4L{\frac {u}{v}}\left(m^{2}-1\right),$

whence by numerical calculation we deduce

$\Delta$ = 0.00010634 millim.

On dividing this difference of path by the length $\lambda$ of an undulation, the magnitude of the displacement becomes

${\frac {\Delta }{\lambda }}=0.2022,$

the observed value being 0.23.

These values are almost identical; and what is more, the difference between observation and calculation may be accounted for with great probability by the presence of the before-mentioned error in estimating the velocity of the water. I proceed to show that the tendency of this error may be assigned, and that analogy permits us to assume that its effect must be very small.

The velocity of the water in each tube was calculated by dividing the volume of water which issued per second from one of the flasks by the sectional area of the tube. But by this method it is only the mean velocity of the water which is determined; in other words, that which would exist provided the several threads of liquid at the centre and near the sides of the tube moved with equal rapidity. It is evident, however, that this cannot be the case; for the resistance opposed by the sides of the tube, acting in a more immediate manner on the neighbouring threads of liquid, tends to diminish their velocity more than it does that of the threads nearer the centre of the tube. The velocity of the