108 Proceedings of the Royal Irish Academy. 
within the indiyidual molecules, and not to the irregular journeys of 
the molecules amongst one another. 
Mr. Stoney thought it possible now to advance another stej) in this 
inquiry, and in the present communication he gave an account to the 
Academy of the grounds upon which he founded this hope. 
A pendidous vibration, according to the meaning which has been 
given to that phrase by Helmholtz, is such a vibration as is executed 
by the simple cycloidal pendulum. It is, accordingly, one in which 
the relation between the displacement of each particle and the time is 
represented by the simple curve of sines, of which the equation is 
y= Cq-\- Ci sin (x-\- a), 
where y - Co is the displacement of the particle from its central posi- 
tion; Ci is the amplitude of the vibration ; x stands for 27r where t 
T 
is the time from a fixed epoch, and t the period of a complete double 
vibration ; and a is a constant depending on the phase of the vibration 
at the instant which is taken as the epoch from which t is measured. 
]^ow we may not assume that the waves impressed on the aether 
by one of the periodic motions within a molecule of a gas are of this 
simple character. "We must expect them to be usually much more 
involved. And whatever may happen to be the intricacy of their 
form near to their origin, they will retain, substantially, the same 
complex character so long as they advance through the open undis- 
persing aether, in which waves of all lengths travel at the same rate. 
But it would seem that a very different state of things must arise when 
the undulation enters a dispersing medium, such as glass. 
Let us suppose that the undulation^' before it enters the glass con- 
sists of plane waves. Then, whatever the form of these waves, the 
relation between the displacement of an element of the aether and the 
time, may be represented by some curve repeated over and over again. 
This curve may be either one continuous curve, or parts of several 
different curves joined on to one another. In the latter case (which 
includes the other) one of the sections of the curve may be represented 
by the equations 
?/ = 0u (^) from ^ = 0 tox = x;i,^ 
3/ = 01 (x) from x = Xito £c = X2, ! ( 1 ) 
and so on, to f 
y = 01 {x) from x = Xiio x = 27r J 
y being the displacement, and x being an abbreviation for 27r ~ 
where t is the complete periodic time of one wave. 
The undulation in vacuo will then be represented, according to 
Pourier's well-known theorem, by the following series : 
y = Aq + cos X K2 cos 2 X + . . . 
+ Bi sin X + B^sin 2 X + . . . 
(2) 
* By the term undulation is to be understood a series of waves. 
