resting upon a Vibrating Support. 57 
responding to various frequencies of vibration, not only in the 
case of water, but also of mercury, alcohol, and other liquids. 
He remarks that the nature of the liquid affects the relation 
in a marked manner, contrary to the theoretical ideas of the 
time, which recognized gravity only as a " motive " for the 
vibrations. In the following year Sir W. Thomson gave the 
complete theory of wave-propagation*, in which it is shown 
that in the case of wave-lengths so short as most of those ex- 
perimented upon by Matthiessen, the influence of cohesion, or 
capillary tension, far outweighs that of gravity. In general, 
if T be the tension, k=1tt /\, the velocity of propagation (y) 
is given by 
»=\/{f +T *} ; (8) 
or, when X is small enough, 
«=V(T*) (9) 
Since X=i'T, the relation between t and X is, by (9), 
2ttTt 2 =X 3 ; (10) 
or, if N be the frequency of vibration, 
N*X= constant (11) 
Dr. Matthiessen's results agree pretty well with (11), much 
better in fact than with the formula proposed by himself. 
There is another point of some interest on which the views 
expressed by Matthiessen call for correction. It was observed 
by Lissajous some years ago, that if two vibrating tuning- 
forks of slightly different pitch are made to touch the surface 
of water, the nearly stationary waves formed midway between 
the sources of disturbance travel slowly towards the graver. 
We may take as the expression for the two progressive waves 
cos (tcx—nt) + cos (fc'x + n't), 
or, which is the same, 
2 cos {^(/e + «/),£ + i(?/— n)t) x cos {\(jd—ic)x + %(n'-+n)t}. 
The position at any time of the crests of the nearly stationary 
waves is given by 
\ (k + k')x + %(n' — n)t = % 
mnr. 
where m is an integer. The velocity of displacement V is 
thus 
v -"^» ™ 
* Phil. Mag. Not. 1871. 
