254 M. H. Fizeau on the Effect of the Motion of a Body 
same manner as before. For the difference of path we have the 
value 
v v 
A=B{— ~— or} 
v— a 
U 
lg 
which by reduction and transformation becomes 
2__ »I2 
=9E" aes 
se 0 Lp (sa) 
ote oer i 
v 
Taking into consideration the smallness of wu with respect to 
fe _ 1 ) perish ’ 
v & = 33000000 )” and the circumstance that the coefficient of 
u? differs little from unity, the term in uv? may, without appre- 
ciable error, be neglected, and the above expression considerably 
simplified. In fact, if m be the index of refraction, and L=3E 
the length of each tube, we have approximately 
A=4L- (m?—1), 
whence by numerical calculation we deduce 
A=0:00010634 millim. 
On dividing this difference of path by the length » of an undu- 
lation, the magnitude of the displacement becomes 
= =0'2022, 
the observed value being 0°23. 
These values are almost identical ; and what is more, the dif- 
ference 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 ana- 
logy permits us to assume that its effect must be very small. 
The velocity of the water in each tube was calculated by divi- 
ding 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 
