Il6 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
like, along a straight line. An equation which satisfies such a periodic 
vibration and which has been found to represent satisfactorily the ether 
disturbances is the following : 
y = a sin (/-/,) (i) 
2irt\ 
in which a represents the amplitude, T the periodic time, - the initial 
phase, / the time which has elapsed at any given instant, and y the distance 
of the ether particle from the position it occupied at the time. 
It can be readily proved* that the intensity of light of a given period of 
vibration (color) varies as the square of its amplitude (a) of vibration. This 
relation is of importance in determining the relative intensities of the plane 
polarized waves which emerge from the upper nicol of the microscope after 
having passed through the lower nicol and an intervening crystal plate in 
different positions. 
Disturbances in the ether which produce light phenomena are ascribed 
to the action of forces on the ether, and if two or more separate disturbances 
are simultaneously impressed on the same element the resultant disturbance 
can be calculated according to the principle of the resolution of forces on 
the assumption of direct superposition of the forces. If, in the case of plane 
polarized light, two separate vibrations be imposed simultaneously on an 
element, the resultant vibration will be in the plane of vibration of the 
components, and its amplitude (on the principle of superposition) will be 
the algebraic sum of the amplitudes of the components. The mathematical 
expression for the resultant vibration of a particle simultaneously impressed 
by two periodic disturbances of the same period, but differing in phase and 
amplitude, can be deduced from the equations of the separate vibrations 
2f 27T 
y\ oi sin (/ /i) and y 2 = a 2 sin (/ / 2 ) 
The resultant displacement at any time / is 
O jr 
2"K f 27T 27T \ 27T / 2T 27T \ 
= sin /( oi cosy/! +Qz cos / 2 ) -cos t ( aisin /i-f-o 2 sin / 2 ) 
-A **(*-$ 
if we consider 
and 
21T 2T . 2T 
cos li+at cos k = A cos t t 
2ir 2ir 2ir 
sin h+ 0j sin <j = A sin / 8 
By squaring and adding the last two expressions we obtain 
s T i) 
Preiton'f Theory of Light, 3d edition, 41-44, 1901. 
