THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 



527 



condensation at any point of the filament is assumed to be (p^ (z, i) xf(.v, y), which for shortness 

 sake, will be written (p^f, (p^ being treated as a function of z and t only, and / as a function of 

 .V and y only. Let p be the density, and ii, v, w the components of the velocity in the directions 

 of the axes of co-ordinates, at the point a;yz, and at the time f. Also let a'-, b'^, c' be the co- 

 efficients of elasticity in the directions of the axes of x, y, ss respectively. First powers only of the 

 velocities u, v, w, and of the condensation p — 1 will be taken account of. This being premised, 

 we have, 



ldu\ a'^dp "''(p, df 



\dt) pd.v p ' dx' 



and to the first approximation. 



du 

 rf7 



- «>. 



dx ' 



He 



,., df , 

 u = - a'--/- f(pdt + C, 

 dx '^ ' 



the arbitrary quantity c being in general a function of x, y, and z. So also 



V = - 6'- ^ f(pdt + C. 

 dy 



Again, since 

 dw 



div 

 ~di 



c"-d 

 odz 



- , we have to the same degree of approximation, 



dt 



= - c'^f- 



dtp, 

 dz 



and 



fd(p^ 



.eyf^dt^c"=.cy'-q^^c". 



J dz dz 



df(pdt 



But from the supposed law of condensation in any plane perpendicular to the axis of z, it follows, 

 that the accelerative force parallel to this axis at any point of the plane, must to the first degree 

 of approximation, be equal to / x the accelerative force at the point of intersection with the axis, 



and the corresponding velocities must be in the same proportion. Hence, — being the velocity 



dz 



at the point of intersection with the axis, we shall have 



w =/ 



d(p 

 dz 



Consequently - c" f(p^dt = (p, and C" = 0. 



Assuming now that C and C' arc each equal to zero when <p = 0, we obtain, 



-" df ....... ^" ^df 



, , and V = ~r,.(t)~. Hence, 

 C • dx c^ ^ dy 





a' df b'' df d(p , 



n dx + vdy + wdz = -^. <p-~- dx + -r ■ <p -^ dy + f ~ dz. 

 c ^ dx c' ^ dy ' dz 



a" h"' 



In this case udx + vdy + wdz is not an exact differential. Let -^^ = /(, and — = ', imd suppose 



ithat/= Fi'.FJ, the function Fi containing x only, and the function F., containing y only. By 

 this supposition, a factor which will render the above quantity an exact differential may be found, 

 which, though not the mo.st general, will suffice for our present purpose. By differentiating, 



