36 



PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



that the direction of motion through a given point remains the same in successive instants. 

 This is rectilinear motion, and it thus appears that the rectilinearity of the motion is in accordance 

 with the dynamical equation.* 



When the motion is perpendicular to a plane, r and r' are each infinite, and equation (8) becomes 



_ This is true whether the motion be in parallel straight lines or in concentric circles about 

 d s 



a fixed axis. But equation (8) does not enable us to determine whether in the latter of these two 

 kinds of motion, udx + vdy + wdx can be an exact differential. This question will be con- 

 sidered further on. 



Reserving then the case just mentioned, the following will be the conclusion to which the 

 foregoing reasoning conducts : — The only motions of an incompressible Jluid which are possible 

 when udx + vdy + wdz is an exact differential of a function of three independent variables, 

 are rectilinear motions. 



8. Now let the fluid be compressible. For the same reason as that adduced in the case 

 of incompressible fluids, equation (9) cannot be integrated between limits entirely arbitrary 

 unless Vp is constant along a given surface of displacement. And again, as before, if s be drawn 

 at a given instant the orthogonal trajectory to surfaces of displacement from any point to 

 a given surface of displacement, then Vp at that point is a function of «. Hence, since 

 o {udw + vdy + wdz) = Vpds, it follows that the left-hand side of this equality is integrable. 

 But by hypothesis udm + vdy + wdx is an exact differential dcf). Hence, since pd<p = Vpds, 

 p is a function of (j), and p and (p are each functions of s. But Vp is a function of s. 

 Therefore V is also a function of s. It is thus shewn that the surfaces of displacement are 

 surfaces both of equal yelocity and equal density. By reasoning precisely as in the case of 

 incompressible fluids a like conclusion is arrived at ; viz. that the only motions of a com- 

 pressible fluid which are possible when udx + vdy + wdz is an exact differential of a function 

 of three independent variables, are rectilinear motions. 



The above result and the analogous one respecting incompressible fluids, are evidently 

 dependent on the fact that when udw + vdy + wdx is an exact differential d<f), both ^ and V 

 are functions of the variable s, which is a line drawn at a given instant in the direction of 

 the motion of the particles through which it passes, commencing at an arbitrary origin and 

 terminating at the point xyss. And again, this fact is a direct consequence from the general 

 equation (3), as may be thus concisely shewn. That equation, on multiplying by N, becomes 



JV -^ + F^ = : or, since V = N. — , it becomes — ^ + iV— -^ = 0. Now when A'' is a function 

 dt ds dt ds 



of t only, and consequently udx + vdy + wdz is integrable of itself, the last equation by 

 integration gives \\, a function of s and t. Therefore also N -j- , or F, is a function of s and t. 

 And since d^ = Vds, <p is also a function of s and t. 



9. It remains to consider what are the forms of the surfaces of displacement which satisfy 

 the condition of rectilinear motion. 



• If all the parts of the fluid have a common motion in 

 a common direction, the surfaces of displacement will partake 

 of this motion, and the motion of the particles in space will 

 not be rectilinear. Such common motions are not the proper 

 subject of consideration in Hydrodynamics. When they exist, 



it must be under given circumstances, and their amount may 

 be calculated in the same manner as for a solid body. These 

 motions may therefore always be considered to he eliminated 

 by impressing equal motions on all the parts of the fluid in a 

 contrary direction. 



