Prof. Chaliis on the Principles of Hydrodynamics. 33 



By satisfying this geometrical condition, the motion of a con- 

 tinuous mass is distingviished from that of a collection of indefi- 

 nitely small discrete atoms. Another general equation is re- 

 quired for expressing that the motion is subject to this limitation, 

 which is not necessarily involved in the equation (2.), because 

 the motion of a collection of discrete atoms might be such as to 

 satisfy that equation. 



Proposition VI. To express by an equation that the directions 

 of motion in any given element are in successive instants normals 

 to continuous surfaces. 



Let i/r be an unknown function of the coordinates and the 

 time such that (f/i/r) = is the differential equation of a surface 

 to which the directions of motion in an element situated at the 

 pomt xyz are normals. By Axiom II. such a sui-face really 



exists. Hence % % ^ are in the proportion oUi, v and 



dx dy dz 

 w. Or, X being another unknown function of the coordinates 

 and the time, 



"=^^I' ^='^^' "='^^- 



Hence 



{df)=^dx+ldy+ ^^dz = 0. 



This equation expresses that the directions of motion m any 

 given element are normals to a continuous surface at one instant. 

 That the motion may be such as to satisfy this condition in suc- 

 cessive instants, it is necessary that the equation 



B{df)=0 



should also be true, the symbol S having reference to change of 

 time and to the change of position of the given element. Un 

 account of the independence of the sjanbols of operation b and 

 d, that equation is equivalent to 



But , . ; , 



and r, cs <>. 



S.r=wS/, By = vBt, 8r = ?fS/. 



Hence 



Phil. May. S. 4. Vol. 1 . No. 1. Jan. 1851. D 



