4 Prof. Powell on the Demonstration of FresnePs Formulas 

 . 11. Here we may observe, in the numerator of Ay, 



2 sin r cos t=: - sin 2t : 

 and of Ap 



28in2r^^' = i2 8in2t. 

 cosr fi 



12. It is also desirable to notice, that these expressions are 

 the same as those given in Mr. Aiiy's Tract, § 129, under the 

 slightly different form in which they directly result from the 

 peculiar process there pursued, viz. writing sin (r — i) and 

 — tan (r— a). 



Also the numerator of k' is positive for all values of (i— r), 

 which is necessarily less than 90°, while the denominator becomes 

 oo at (t + r) = 90°, which, according to Brewster's law, is the 

 polarizing angle, and for greater values continues negative. 



13. Prof. Maccullagh's formulas are, — 



for vibrations parallel to the plane of incidence, 



jj_ Bin (i-r) ,_ sin2t , .„,. 



'*-" sin (i + r)' '''~sin(f + r)' ^^ 



for vibrations perpendicular to the plane of incidence, 

 ,f__ sin2i— sin2r __tan (e— r) 

 ~" sin 2 j -f sin 2r ~ tan {i -f r)' 



2sin2i _| tan(i — r) .^,. 



'"" 6in2i + sin2r tan(i+rj* 



These last are sometimes expressed under the forms 



Li __ s^^ (* "■ ^) ^^^ (* + ^) z. sin 2i 



"" cos (j—r) sin (t + r)' '~ cos {i—r) sin {i-\-r) ' 



14. Comparing these formulas with FresnePs (distinguished 

 by using roman letters), we may observe from (11), 



A'=-h', h,=h,fji, 



kf=k', k,=k,fi. 



Here also k, undergoes the same change of sign at the incidence 

 of complete polarization. 



Densities and Vibrating Masses. 



16. In deducing these formulas, it is in all cases necessary to 

 express the ratio of the masses of sether simultaneously vibrating 

 without and within the medium; and the differences in the 

 respective formulas are mainly dependent on the very opposite 

 suppositions made by the several philosophers as to the density 

 of the sether in different media, — Fresnel supposing it more dense 



