Endosmose and other Allied Phenomena. 69 



Substituting the values of dwjdx, dw/dy from (48), we see 

 that if we neglect IA, IB, IG in comparison with unity, we 

 have 



-.du 



U=lj-, 



% > 



Hence, if q denote the velocity parallel to a tangent-line at any 

 point P of the wall, we have at P 



■-'£ < 52 > 



or, if \j, fa, Vi be the direction-cosines of the normal, and 

 \ 2 , fi 2 , v 2 those of the tangent-line, 



\ 2 u + /n 2 v + v 2 w = I f\ - — 1-^^+ v x j- J {\ 2 u + fi 2 v + v 2 io), (53) 



in which of course \ 2 , fi 2 , v 2 are to be treated as constants 

 during the differentiations. Let us apply this to the case 

 when P is any point (x, y) near the origin. The values 

 of \ x , fi^ v x for this case have been given in (47), whilst we 

 may write 



X 2 : /n 2 : v 2 = dx : dy : (A# + By)dx + (Bx -f Cy)dy + . . . 



Substituting in (53), and equating coefficients of dx, dy, we 

 find 



, . -r, ,/ dw dw div \ ~) \ 



+ (A-+^r+...)(». s +*.,^ + ii ar ;j, 



._, „. N A ^ dv , dv , dv 



v +( Bx + C y + '-> = l {^dx + ^dy +Vl dz 



+(B.+c!,+...)(x 1 . 5 +fc^+i> as -;j. 



In these equations m, u, w, &c, denote the values of these 

 quantities at the point (x, y), and must be expanded in terms 

 of x, y. Performing the expansions, and equating coefficients 

 of x, y, we get the following four relations : — 



V. (54) 



