AND REFRACTION OF LIGHT. 9 



Let us now consider the particular case of two indefinitely extended 

 media, the surface of junction when in equilibrium being a plane of 

 infinite extent, horizontal (suppose), and which we shall take as that 

 of (yz), and conceive the axis of x positive directed downwards. Then 

 if p be the constant density of the upper, and p t that of the lower 

 medium, 2 and 2 " the corresponding functions due to the molecular 

 actions. The equation (2) adapted to the present case will become 



fffp dx dy d% tgp Su + -~ lt> + -^ lw\ 



+ fffp. d * d y d % \jf * «, + rfjr *•, + -$■ 3 ™ j , (3). 



= fff dx d V d% & + fffdx dy dz <$> ; 



u t , v t , w t belonging to the lower fluid, and the triple integrals being 

 extended over the whole volume of the fluids to which they respectively 

 belong. 



It now only remains to substitute for 2 and ffi their values, to effect 

 the integrations by parts, and to equate separately to zero the coefficients 

 of the independent variations. Substituting therefore for <p 2 its value (C), 

 we get 



fffdx dy dx S(p 2 

 . rrr , , » f (du dv dw\ [d$u d$v d$w\ ) 



= - A ^ dxd y d% \{Tx + Ty + ^)\^ + ~^ + -d^)) 



J? /7T/7 // // \(dw dv\ (dlu d$v\ (du dw\ ld§u d$w\ 

 •'•'•' * \\dy dx) \dy dx) \d% dx) \d% dx I 



(dv dw\ (dSv d§w\ r/rf» d$w dw dly \ 



\d% dy) \d% dy) L \dy d% d%' dy ) 



(du d§w dw d%u\ (du d$v dv dSu \ -i 1 

 \dx' d% d%' dx] \dx dy dy' dx/jj 



Vol. VII. Paet I. B 



