AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 299 



{du idu dv\ idii dw\\ 



dx \dy dxl \dz dx I ] 



dL dL dL 



in which equations I, m, n will have to be replaced by — , - , ^ , to which they are pro- 



\X CL u If (Ji *j 



portional. 



If we choose to take account of capillary attraction, we have only to diminish the pressure 11 



by the quantity H ( 1- ~J> "here H is a positive constant depending on the nature of the fluid, 



and r,, r, are the principal radii of curvature at the point considered, reckoned positive when 

 the fluid is concave outwards. Equations (14) with the ordinary equation 



dL dL dL dL 



37 + " 3— + "-5- + ""^T" = °' ('5) 



dt dx ay dz 



are the conditions to be satisfied for points at the free surface. Equations (14) are for such 

 points what the three equations of motion are for internal points, and (15) is for the former 

 what (11) is for the latter, expressing in fact that there is no generation or destruction of fluid 

 at the free surface. 



The equations (14) admit of being difiierently expressed, in a way which may sometimes 

 be useful. If we suppose the origin to be in the point considered, and the axis of ss to be the 

 external normal to the surface, we have I = m = 0, w = l, and the equations become 



dw du dw dv „ dw 



:^+^ = o, —+-- = 0, n -p + 2,ji— = (16) 



dx dz dy dz dz 



The relative velocity parallel to 2; of a point (so', y, 0) in the free surface, indefinitely near 



. . . dw , dw , ^ ^ dw dw , , , . . , , 



the origin, is - — x + -p- y : hence we see that — , — are the angular velocities, reckoned 

 dx dy dx dy 



from X lo z and from y to z respectively, of an element of the free surface. Subtracting the 

 linear velocities due to these angular velocities from the relative velocities of the point {x, y', z'), 

 and calling the remaining relative velocities U, V, W, we shall have 



du , du , (du dw\ , 

 U = —X + —y + — +3— *' 

 dx dy \dz dx I 



dv , dv I fdv dw\ , 

 dx dy \dz 



dy \dz dy I 



n = — z . 

 dz 



dU 

 Hence we see that the first two of equations (16) express the conditions that - , = 



dV 



and — ; = 0, which are evidently the conditions to be satisfied in order that there may be no 

 dz 



gliding motion in a direction parallel to the free surface. It would be easy to prove that these 



are the conditions to be satisfied in order that the axis of z may be an axis of extension. 



The next case to consider is that of a fluid in contact with a solid. The condition which first 



occurred to me to assume for this case was, that the film of fluid immediately in contact with the 



solid did not move relatively to the surface of the solid. I was led to try this condition from the 



following considerations. According to the hypotheses adopted, if there was a very large relative 



a Q 2 



