resting upon a Vibrating Support. 53 
The current was somewhat in excess; so that the desired speed 
could be attained by the application of moderate friction. At 
a certain speed of rotation the appearances were as follows. 
Through the set of four holes (giving four views for each 
rotation of the disk) the bar was seen double. Through the 
set of two holes the bar was seen single, and the water-waves 
were seen double. Through the single hole the bar was seen 
single, and the waves also were seen single. From this it 
follows that the water vibrations are not, as Matthiessen con- 
tends, synchronous with those of the bar, but that there are 
two complete vibrations of the support for each complete vi- 
bration of the water, in accordance with Faraday's original 
statement. 
An attempt was made to calculate the frequency of liquid 
vibration from measurements of the wave-length and of the 
depth. The depth (h), deduced from the area of the plate and 
the whole quantity of liquid, was "0681 centim.; and by direct 
measurement X=-848 centim. Sir W. Thomson's formula 
connecting the velocity of propagation with the wave-length, 
when the effect of surface-tension is included, is 
t 2 V277 A. / e a + e~ a ' v y 
where a = 27rh/X. With the above data we find for the fre- 
quency of vibration (t -1 ) 208. This should have been 15 - 5; 
and the discrepancy is probably to be attributed to friction, 
whose influence must be to diminish the efficient depth, and 
may easily rise to importance when the total depth is so small. 
Another method by which I succeeded in determining the 
frequency of these waves requires a little preliminary expla- 
nation. If w=2tt/tj and k=2tt/X j the stationary waves 
parallel to y may be expressed as the resultant of opposite 
progressive waves in the form 
cos (/e# + 7tf ) + cos (jcx— nt) = 2 cos kx cos nt. . . (1) 
This represents the state of things referred to an origin 
fixed in space. But now let us refer it to an origin moviDg 
forward with the velocity (»/«) of the progressive waves, so as 
to obtain the appearance that would be presented to the eye, 
or to the photographic camera, carried forward in this manner. 
"Writing kx 1 + nt for kx, we get 
cos {jcx 1 + 2ni) + cos kx / (2) 
Now the average effect of the first term is independent of af\ 
so that w r hat is seen is simply that set of progressive waves 
w 7 hich moves with the eve. In this wav a kind of resolution 
