30 Prof. Challis on the Principles of Hydrodynamics. 



Proposition III. To obtain the general equation of the equill 

 bmim of a tiuid. 



Let the coordinates of the position of any element of the fluids 

 referred to three rectangular axes of coordinates, be x, y, z, 

 which we may for the sake of simplicity suppose to be all posi- 

 tive, since the results obtained by reasoning throughout con- 

 sistently with this restriction will be true independently of the 

 restriction. Let the form of the element be that of a rectangular 

 parallelopipedon, and its edges dx, dy, dz be parallel to the axes 

 of coordinates ; and let p be its density. Then if X, Y, Z be 

 the impressed accelcrative forces acting on the element in direc- 

 tions respectively parallel to the axes of coordinates, and tending 

 from the coordinate planes, the impressed mo\-ing forces in the 

 same directions are 



X/3 dx dy dz, Yp dx dy dz, Zp dx dy dz. 



These are counteracted by the excesses of the pressures on th« 

 faces of the element fm-thest from the origin above the pressures 

 on the opposite faces, het pdydz, qdxdz, rdxdy he the ])vess\ivea 

 acting respectively parallel to the axes of x, y, z on the faces 

 nearest the origin. Then the excesses of pressure tending to- 

 wards the coordinate planes are 



-j-dxdydz, -j^dxdydz, -j-dx dydz. 



But by Proposition I., p, q, r differ fi-om each other by infinite 

 simal quantities. Hence substituting p for q and r, and equa 

 ting these pressures to the impressed moving forces acting in the 

 opposite du'ections, the resulting equations are 



Hence, since 

 we have 



^=X -^ =Y -^ = Z 



pdx ^ ' pdy ' pdz 



{dp)=^£dx+^^dy+f^dz, 

 ^^=Xdx + Ydy + Zdz. 



This is the required equation, which, being proved of any ele- 

 ment, may be extended to all. The above, which is the usual 

 investigation of this equation, is introduced here for the sake of 

 exhibiting completely the reasoning by which the fundamental 

 equations of hydrostatics and hydrodjaiamics are established. 



Proposition IV. To obtain a general dynamical equation ap- 

 plicable to the motion of a fluid. 



Let X, y, z be the coordinates of the position of any element 



