iJi Prof. Powell on the Demonstration of PresnePs Formulas 



are here concerned only with the ratios of the amplitudes, we 

 may suppose one of them the same as in the last case, and thence 

 find the others, which will be different in order to fulfil the dif- 

 ferent conditions. Thus assuming k,=hi, the condition of equi- 

 valence (L) gives 



/A . jLh 1 cosr ^ . ^.cosr ... 



(n-\-h')=h. r=3sm3f ;=4smtco8r : 



' cost cos I 



and from this, with that of vis viva (M), viz. 



cos r 

 {h^ — h'^) = h^ r = 1 6 sin i cos i sin r cos r = (A + h') (h — h!) 



(A—//) =4 cos i sin r 



2A = 4 sin i cos r + 4 cos i sin r 



2h' = 4 sin t cos r — 4 cos i sin r ; 



and since always sin i > sin r and cos r > cos «, the last value is 

 always positive. Hence dividing by h, considered = 1, 



hi^'Mz^ _^sin2f . . . (H) 



the same values as those derived from the geometrical construc- 

 tion before mentioned (24). 



41. '(II.) On the hypothesis of increased density (or of in- 

 creased retardation supposed to be represented by the same law), 

 we have for the masses 7w = sinrcos«, m^=sinicosr, in which, 

 since sin r < sin i and cos i <cos r, we have always m < m^, and 

 therefore when we substitute in equation (N), we have 



m—mi^ —(sin i cos r— sin r cos i)= — sin (t— r). 

 Thus from that equation we have directly, dividing by h, as before, 

 ,, — sin (i— 7') , 2 sin r cos i ,j,. 



sm {t + r) ' sm {t + r) ^ ' 



These are PresnePs formulas as derived originally from assu- 

 ming the same equation (N) on the analogy of impact. 



42. But it is also evident from the process of elimination by 

 which that equation is here obtained, that these values fulfil the 

 equation of vis viva, 



(h^^h'^)=zh,^fi^^ = sin 2i sin 2r, 

 cos I 



as well as that of equivalence of vibrations (L), 



h-\-h/=h,fOT sin (i + r) — sin (i — r) c= 2 sin r cos i ; 



which last proves that they necessarily belong to vibrations per- 

 pendicular to the plane of incidence. 



43. Por vibrations parallel to the plane of incidence: — as 



