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(which includes, as a particular case, the remarkable law for deter- 
mining the angle of polarisation of light reflected at the surface of 
an ordinary medium, discovered by Sir David Brewster,) namely, 
that when the light reflected from the surface of a doubly refract- 
ing crystal is completely polarised, or, in other words, when the 
reflected vibration has a determined direction, independent of the 
direction of the incident vibration, then the reflected ray is per- 
pendicular to the intersection of the polar planes of the two dif- 
Serent refracted rays. 
In this and other applications of the theorem of the polar plane 
to the case where the incident light is polarised so as to undergo a 
double refraction, the obvious manner of proceeding is to decompose 
its one biradial vibration into two uniradial vibrations, and to treat 
these separately, by applying to each the construction above described. 
Yet Mr. Mac Cullagh remarks, that it requires proof that the reflected 
and refracted intensities, thus determined, will have their sum ex- 
actly equal to the intensity of the incident light ; or, in other words, 
that the law of the vis viva will hold good for the resultant vibra- 
tions, though we know, by the construction, that it holds good for 
each system of uniradial components taken separately. In fact, if the 
two separate incident vibrations, which correspond to the two sepa- 
rate refracted vibrations, be inclined at an acute angle to each other, 
they will generate by their superposition (according to the law of 
interference) a compound incident light, of which the intensity ex- 
ceeds, by a determined amount, the sum of the two separate or 
component intensities ; and it requires proof that the two separate 
reflected vibrations will in like manner be inclined to each other at 
that precise acuteness of angle which will allow the intensity of the 
compound reflected light to exceed, by precisely the same deter- 
mined amount, the sum of the two separate intensities, correspond- 
ing to the two separate reflected vibrations: (or that the same sort 
of equality of differences between incident and reflected resultants 
and sums will take place, when the angles are obtuse and not acute ;) 
the two refracted vibrations being not in general (in either case) 
superposed upon each other. Professor Mac Cullagh has arrived 
at an equation of condition, as necessary for the foregoing agree- 
ment, which expresses a property of the laws of propagation de- 
