6 Prof. Powell on the Demonstration of FresnePs Formulas 



27. But these are merely particular cases of the general theory 

 of "equivalence of vibrations/' first, I believe, systematically 

 proposed by Prof. Maccullagli as the basis of his higher investi- 

 gation of the laws of reflexion and refraction at the surfaces of 

 crystals. Yet some cases of it appear to have been assumed by 

 Fresnel, though under a slight, but material, difference of view. 

 The general principle common to both is, that in " two conti- 

 guous media, the incident, reflected, and refracted vibrations are 

 mechanically equivalent : " but a difference in conception of the 

 distribution of the force among them gives rise to a corresponding 

 difference in the form of the expressions ; on either view, how- 

 ever, these expressions indicate conditions distinctively applying 

 to vibrations respectively parallel and perpendicular to the plane 

 of incidence. 



28. Taking three co-ordinate planes, XY that of the surface, 

 XZ that of incidence, and YZ perpendicular to incidence, any 

 vibration h passing through the origin taken at the point of in- 

 cidence, and inclined to XZ by an angle 0, and to XY by <^, 

 may be resolved into 



y=:hsmO, a?=Acos^cos</>, z=hcos6sin(l>. 



Then for h of the incident ray we have <^=:i and 9 



A' of the reflected ray we have </>= 2 and 6' 



• . • hfO{ the refracted ray we have = r and O^. 



The law of equivalent vibrations, according to Prof. Mac- 

 cuUagh, is expressed by these relations between the resolved 

 parts respectively : — 



in z, h cos 6 sin i + U cos 6' sin i = h^ cos 6^ sin r ; 



in y, A sin ^ + h! sin &=^h^ sin 6^ ; 



in a: J h cos 6 cos i + h! cos 6^ cos i = h^ cos 6^ cos r. 



29. Hence if the plane of vibration coincide with YZ perpen- 

 dicular to incidence, ^=^' = ^^=90°, and the law becomes simply 



If it coincide with XZ, 0=0'=6f = 0, and it becomes 

 h cos i H- h' cos i = hj cos r, 



A + A'=A,'5iU(/4). 

 'cos 7 ^ '' 



These two may be included in the formula 



(A,)=A,(sin^ + ^cos^). . . . (L) 



5=90° gives (h,)=hi, and 6=0 gives (k,)=h,^^. 



