194 UNDULATORY THEORY OF LIGHT. 



copcc. The living force in the reflected beam (which we may represent bj E) 

 will consequently be — the mass being assumed for convemeuce=unity — 



11^ ri^'h'.co.^.+ C'^\':^\)'.lu^a. [27.] 



The agreement of this result with our previous conclusions may be verified 

 by making a succe-sively;:=0-' and =90'^. In the first case — 



11= / . —-A ; and m the second, R= I — > ^^ j 



If a =45°, sin^a=^, and cos"a=:^i. Couscqucntly» 



This might easily be anticipated by considering that, in the supposed case, 

 the incident beam is equivalent to two beams, each having an intensity of ^, 

 and polarized — one in the piano of reflection, and the other in the plane perpen- 

 dicular to it. The reflected beam should contain one-half the force in each 

 plane which it would have done had each intensity been = 1. 



Let there now be two beams each =^h incident together, and polarized in thu 

 azimuths a and a'. From what has just been said, it is evident that the value 

 of R will be— 



ll = i("4^)^cos^«+cos="'; + ^(''"!'-'1)'.(Bin^. + BinV). [29.] 



In this expression, if u'=::^90^ — a, cos"a-|-cos'a'=:l. Also sin^a-j-sin^a'::=l. 

 R becomes, therefore, equal to the sum of the intensities of two rays each =^, 

 polarized, one in the plane of incidence, and the other at right angles to it, no 

 matter what may be the value of a. If, then, any number of waves, in different 

 azimuths, follow each other in so close succession as to blend their impressions 

 upon the eye, and if their azimuths are so impartially distributed that lor every 

 value of a there is another = 90^ — a, the forces in all these azimuths being 

 equal, then the resultant effect of the whole must necessarily be — 



''-<\ni.+r)) +^tan(= + ,); 



[30. 



But this is the condition of common light. The formula just stated, there- 

 fore, represents the living forces in the two principal planes, in a beam of com- 

 mon light after reflection, the original force being taken = 1. When c-\-p=z90'\ 

 the second term disappears. The reflected 'beam is then entirely polarized. 



If we decompose the second term in the value of R, above, into its factors, 

 we shall have (disregarding the numerical coefficient, and omitting the expo- 

 nent) — 



tiXM{c-f >)^sm{'.-p) cos(M:r) rg-^ j 



tnn[t-j-p) sin(!-f-/>) "cos(£— /j) * 



The molecular velocity of the wave polarized at right angles to the plane of 

 reflection appears thus to be equal to that of the wave polarized ?n the plane of 



reflection, multiplied by the factor — -,-^-^-^~.- When £=^90° (which is the great- 

 ^ '' cos{c—p) ^ '^ 



est value it can have) the numerator and denominator of this fraction are equal, 



with opposite signs. The sign does not concern us at present, as it has no effect 



upon the value of the living force in the wave. For all values of i less than 90"^ 



[f) being necessarily less than t when n exceeds 1) the denominator is greater 



than the numerator. It follows that in the reflection of common light, a larger 



amount of living forco will, in the reflected beam, be preserved in the movements 



