Stoney — On the Interrupted Spectra of Gases. 
109 
where the coefficients are obtained from equations ( 1 ) by the definite 
integrals. 
I y cos nx, dx= tt A^. 
y sin nx, dx=7r B^. 
Jo 
Equation (2), the equation of the undulation before it enters the 
glass, may be put into the more convenient form 
y -Aq = Ci sin (x + a^) + Ci sin (2x + a^) + . . . (4) 
where y -AqIS the displacement from the position of rest, and the new 
constants are related to those of equ. (2) as follows : 
A,, 
Cn = V An' + i^n^ «« = tan-\ ^. (5) 
The first term of expansion (4) represents a pendulous vibration 
of the full period t : the remaining terms represent harmonics of this 
vibration, i. e. their periodic times are |t, &c. All of these also 
are pendulous, so that equ. (4) is equivalent to the statement that 
whatever be the form of the plane undulation before entering the glass, 
it may be regarded as formed by the superposition of a number of 
simple pendulous vibrations, one of which has the full periodic time t, 
while the others are harmonics of this vibration. 
Moreover, these vibrations will co-exist in a state of mechanical 
independence of one another, if the disturbance be not too violent for the 
legitimate employment of the principle of the superposition of small 
motions. So long as the light traverses undispersing space these con- 
stituent vibrations will strictly accompany one another, since in open 
space waves of all periods travel at the same velocity. The general 
resulting undulation will, therefore, here retain whatever complicated 
form it may have had at first. Eut when the undulation enters such 
a medium as glass, in which waves of different periods travel at dif- 
ferent rates, the constituent vibrations are no longer able to keep 
together, each being forced to advance through the glass at a speed de- 
pending on its periodic time. Thus, there arises a physical resolution 
within the glass of series (4) into its constituent terms. And if the 
* Other expansions similar to Fourier's series can be conceived, in which the terms, 
instead of representing pendulous vibrations, should represent vibrations of any other 
prescribed form ; and hence, a doubt may arise whether the physical resolution effected 
by the prism is into the terms of the simpler series. That it is so may, perhaps, not be 
susceptible of demonstration ; but the following considerations seem to show it to be 
probable in so high a degree that it is the hypothesis which we ought, provisionally, to 
accept. For, firstly, the form of the emerging vibrations is independent of the material 
of the prism, since the lines correspond to the same wave-lengths as seen in all prisms; 
and, secondly, it is independent of the amplitude of the vibration within very wide 
limits, since the positions of the lines remain fixed through great ranges of temperature, 
and, in many cases, when the temperature falls so low that the lines fade out through 
excessive faintness. 
