108 Prof. Powell on the Demonstration of FresnePs Formulas 



general its plane of polarization is changed', at incidences less than 

 that of complete polarization, the new plane of polarization de- 

 viates on the side of the plane of incidence ojjposite to that of the 

 original plane of polarization ; at the incidence of complete po- 

 larization it coincides with the plane of incidence ; at incidences 

 greater than that of complete polarization it deviates on the same 

 side as the original plane. 



9. Now these are precisely the changes indicated by a very 

 simple deduction of theory, derived from the original formulas of 

 Fresnel. The deduction is well known, but it will be desirable 

 to exhibit its nature explicitly as follows : — 



Let the original plane of polarization (P) be inclined to that 

 of incidence (1) by an angle (a), then after reflexion the parts of 

 the amplitude resolved parallel and perpendicular to (I), viz. — 



J . tan (i—r) , —sin ii—r) 



K sin a= . ^ \' . \ sm a and K cos a= — : — -^ — ^' cos a, 

 ±tan(? + r) sm(2 + r) 



will by composition give a resultant ray polarized in a plane (Q) 

 inclined to (I) by an angle (/S), where 



*-(«) 



tany£^=l ■^, Itana. 



Hence at first, the tangents having opposite signs, the arcs u 

 and yS lie in adjacent quadrants; at i + r=90°/3=0; and for 

 incidences greater, the tangents having the same sign, a and /3 

 lie in the same quadrant, which exactly expresses the experimental 

 results. 



10. It should be remarked that this is the reasoning adopted 

 by Mr. Airy (§ 132), and that the undeniable conclusion follo>vs 

 simply by virtue of the symbols , without the introduction of any 

 extraneous construction or subsidiary consideration whatever. 



11. Now Dr. Lloyd (in his 'Lectures on the Wave Theory,' 

 part 2, p. 35), after stating the facts, gives the same deduction 

 from theory, using MaccullagUs formulas ; but with due caution 

 makes the inference only so far as to show the existence of a devia- 

 tion to the same amount, without eocpressing in which direction. But 

 according to these formulas, tan /3 is at first positive, and changes 

 to negative at the polarizing incidence. Hence as to the direction 

 of the deviation, if the formei' reasoning be correct, then in this 

 case, on the same grounds, the symbols would indicate that the 

 deviation at incidences less than that of polarization must be on 

 the same side as the original plane, and after that incidence, on 

 the ojyposite side ; which is the reverse of the former conclusion, 

 and of the fact, 



12. This, however, would appear to be only an additional 

 reason to that already assigned for the rejection of Maccullagh's 



