268 Prof. Powell on the Demonstration of PresnePs Formulas 



be this physical value, we must take 



(A,)=A,i=((A) + (A'))l 



Thus both cases of equivalence may be included in one expression 

 {0 being the inclination of the plane of vibration to that of in- 

 cidence), ^^^^ _^ ^^^,^^ ^^^ + cos ^ i ) = (h,), 



where ^=90° gives case (a), and ^=0° gives case {j3). 



7. If this be admitted as the true expression of the law of 

 equivalence, we can deduce both FresneVs original formulas from 

 this law of equivalence combined with that of vis viva, viz. 



{h^'-h"')m=hi^m,. 



(a) For vibrations perpendicular to the plane of incidence, the 

 law of equivalence, {h + h') = h,, combined with the law of vis 

 viva, gives 



(h--h')^:=^=h + h. 



But on the hypothesis of increased density or retardation, as 

 before (first paper, § 17), 



m __ sin r cos i 



^■"sinecosr ^^^^ ^^\f^S> ^;' 



(h — h') sin r cos i = ( A + A') sin i cos r, , avodfj 



or 



A (sin r cos i — sin i cos r) = A' (sin r cos i -\- sin i cos r) 



^fii mo'fl ^I:*^ _ -(sin(i-r)) ^ ^ 2 sin r cos a )HT 



A sin(2 + r) ' h sin(2 + r)' iJsufK^ 



8. (/3) For vibrations parallel to the plane of incidence, the law 

 of equivalence (writing kh!k^ instead of [h) {h') {h^), 



ai li 6B omac oxiJ my i ^^^-'^ t.'-^^ ^^ J-^ -f-^ 



(£ni»3 (B) OKBO ni iud , (^ + k') - = A:^, .^qua b^d aw (»)^ 68f>) 



^ .eaiiisnab 



coiftbiittd wiyi tfesilawftf t?i« vimy giveavisedo ad oeIb x^m il 

 9jdi ni i(lfio. oJ boj^o'l^'i a a! oiu| an to yi<(^f>aB adJ iIs^IIjjodbM bnB 

 3r>dw ^93ii*ini^i5r^ffiS^^fc^<(i'f^% 98bo 



•dhf^.^iq j>ili ill ^iiuf^ii^/ rM(iD'iij«j(|« Jofi^i Y:qohjaii iBoiiifiibam sdt 

 "^^ ■ . . ! p. I ^ J } 993 3W floilBsiigyvni 



;«_ , ^ smrcose ^ sm^ ^ sin2j ^ ^ ^^^^^^^^^ anoitBidi'/ 



m, sm e cos r sin^r ^ smgndo^ni ^Bnibto nf ssb') 



(yfc-AO sin 2i= (AdrJ') sin.2r. 



