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CHAPTER IV. 



PRINCIPLES OF HYDROSTATICS. 



32. Pressure, Isobaric Surfaces, and Gradient. The theory of pressure as 

 met with under general conditions in strained elastic bodies or in moving viscous 

 fluids is of great complexity. But in the special case of the equilibrium of any 

 fluid, as well as in the case of the motion of a frictionless fluid, it is reduced simply 

 to a scalar quantity. The field of a hydrostatic pressure can therefore be described 

 according to the common principles for the description of scalar fields (section 

 16). Thus for the geometric representation of this field we draw surfaces of equal 

 value of pressure, or isobaric surfaces. As a rule we shall draw them for unit- 

 differences, so that they divide the space into a set of isobaric unit-sheets. To get 

 unit-sheets of the proper thickness we are free to choose a unit-pressure of suitable 

 magnitude. 



The -pressure gradient, or simply the gradient G, is given by the rate of 

 decrease of the pressure p along the normal n to the isobaric surfaces 



W G= ~Tn 



and the component G s of the gradient along any direction s is given by the rate of 

 decrease of the pressure along this direction 



G = ~ 8 i 



The isobaric surfaces and the unit-sheets, drawn for a unit of suitable magnitude, 

 give the full representation of the field of the gradient G (section 17). The vector 

 itself is directed along the normal to the surfaces, its numerical value being equal to 

 the reciprocal thickness of the sheet. Its component G s along any line is equal to 

 the reciprocal length of that segment 5 of this line which is contained in the unit- 

 sheet. In accordance with formula (f), section 17, we have finally 



\ c ) Cr,= " 



which gives the mean value along the curve i- of the component of the gradient 

 tangential to the curve. The mean tangential gradient is thus equal to the differ- 

 ence of pressure at the end-points of the curve, divided by the length of the curve. 



33. Dynamic Significance of the Pressure Gradient. Like every scalar 

 quantity, pressure has a gradient. But the gradient of the pressure has at the same 

 time a dynamical significance, making it the fundamental vector of hydrostatics and 

 hydrodynamics. 



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