u Regula Tertia Philosophandi." 27 



In proof of principle (1) it suffices to say that a finite change 

 of direction of the motion of an element in an infinitely small 

 interval could only be produced by infinite force, which, by 

 the nature of the inquiry, is excluded. The principle (2) of 

 geometrical continuity, being supposed independent of change 

 of time and place, requires to be expressed as such, like that 

 of constancy of mass, by a differential equation, the mathe- 

 matical investigation of which I proceed now to give. 



If u, v, w be the velocities at the point xyz at the time t in 

 the directions of the axes of x, y, z, it is known that in 

 the case supposed of surfaces of displacement of continuous 

 curvature, udx + vdy + wdz is either integrable of itself or by 



a factor. Hence if - be the factor, when one is required, and 



Ai 



(dyfr) be put for the exact differential -dx+-dy+— dz, we 



shall have (dijr)=Q for a surface of displacement. The above 

 stated principle of continuity requires that such an equation 

 should apply to each element of the fluid at successive instants. 

 This condition is expressed by the formula (d\lr) + 8(dyjr) — 0, 

 the sign of variation 8 having reference to change of position 

 of the element in space and time. Hence as (dyjr) = 0, we 

 shall have also S(dyfr) = 0, and, on account of the independ- 

 ence of the signs of operation S and d, (d . S\fr) = 0, whence by 

 integration 8^ = cj)(t)8t. The complete variation Syjr of the 

 function yjr in the time St is, by the Calculus of Variations, 



yfr being a function of x, y, z, and t. Since the variations 8x, 



Sy, Sz apply to the change of position of the given particle in 



the small interval St, Sx = uSt, Sy = v8t, and 8z = w8t. Also 



, u d-^r v dylr w d-^r TT , . ... . 



we have r = -r- 1 =r = ^S^r= i • Hence by substitution, 

 X dx 1 X dy ' \ dz J ' 



and, after rejecting the common factor St, supposing </>(£) to 



be included in -~, we obtain 

 dt 



This is the required equation of continuity. By this equation 

 the mathematical theory of the motion of a fluid is completed, 

 inasmuch as the three differential equations, together with the 



equations u = \-j-^, v=\-~, w = \-¥-j and a given relation 



between the pressure p and the density p, furnish seven equa- 



