106 Prof. Powell on the Demonstration of FresnePs Formulas 



VII. FresnePs hypothesis of vibrations perpendicular to the 

 plane of polarization. 



3. By different combinations of these principles, different 

 modifications of the formulas result. Thus wc have the hypo- 

 theses — 



(A) combining Nos. I. II. IV. VI., whence are obtained the 

 formulas 



h'= ~ — )-. i .... h,^— — r- r .... vibrations parallel. 



sm (i -,'- r) ' sin (i + r) ^ 



which are Maccullagh's formulas ; the double sign indicating the 

 change at the polarizing angle. 



(B) Combining Nos. I. II. V. VII., whence are obtained, 



,, — sin(i— r) . 2 sin r cos i ,. , 



n'= — ^ — r^ — r-^ n,= —. — TT — T— .... perpendicular, 

 sin (i + r) ' sm (e + r) 



^ — tan(i— ?') / tan(i— r) \cos2 „, 



k!= -— — ;., [ k.= [ 1 , , \. . \ I parallel. 



+ tan (t + r) ' \ ± tan (i-\-r)/ cos r ^ 



(C) Combining Nos. I. III. V. VII., whence are obtained, 



,, sin (z— r) , 2 sin r cos i ,. , 



A'= — — TT--— ^ n,=: —. — -r. r- .... perpendicular, 



sin(« + ^) sin(2+r) ^ ^ 



^_ tan(z— r) ^^_A_ tan(i-fr) \ cosi 



±tan {i + r) ' \ ±tan (z-f-r)/ cosr * ' * * *^ 



4. Each of these two last sets differs from FresneFs in the 

 si^ns. FresnePs original formulas can only be produced from 

 assuming hypothesis (B) for ^, and (C) for k ; or we have, — 



(D) combining Nos. I. Ila. IllyS. V. and VII., whence are 

 obtained, 



jj__ —sin (i—r) , _ 2 sin r cos i 

 8in(i + r) ' sin (f + r) * 

 tan [i—r) , _ / ^ tan {i—r) \ cos i 



k'= r ',. \ k^ 



-(}-i 



±tan(24-r) ' \ + tan (z-fr)/cosr ' 

 which are FresnePs original formulas. 



5. With regard to the law of equivalent vibrations, it may 

 indeed be remarked that Prof. Maccullagh in stating it, with a 

 view to his ulterior researches on crystalline reflexion, rather 

 assumes than demonstrates the main principle, and thus the mo- 

 dified form of that law (No. III.) may possibly be as open to 

 consideration as the original form. But as neither form exclu- 

 sively will produce FresnePs original formulas, it becomes of 

 more importance to look to some other principle which might 



