172 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY 



of the atmosphere ; through the point D draw the straight line 11 1 

 parallel to the surface of the sea, and HI will manifestly be the 

 asymptote to the curve described by the extremities of the diameter. 



Put r=ac, the radius of the globe at the bottom of the water, 

 d cc, the depth of the water at the place of immersion, 

 7i CD, the height of a column of water equal to the weight of 



the atmosphere, 

 x rr CG, any abscissa, 

 y nz G E, the corresponding ordinate, or radius of the globe. 



Then, because the magnitude of the globular body, is inversely as 

 the density, (the weight and the quantity of matter remaining the 

 same,) and the density is directly as the pressure ; it follows, that the 

 magnitude of the body at different points of its ascent, is inversely as 

 the pressure at those points, and the pressure is directly as the depth ; 

 therefore, we have 



DC : DG : : GE S : ca 8 ; 

 but according to the foregoing notation, we have 



d+h : h + x:: y* : r 3 ; 



from which, by equating the products of the extremes and means, 

 we get 



hence, by division, we obtain 



= 



(*+*) 



and by extracting the cube root, it is 

 ,/(d + h) 



r y (T+T)' (133). 



The equation in its present form, exhibits the nature of the curve 

 described by the diameter of the body during its ascent; or it ex- 

 presses generally, the value of the ordinate or radius corresponding 

 to any depth ; but in order to determine the radius at the surface, 

 which is the primary demand of the problem, we must suppose the 

 quantity x to vanish, in which case, the above equation becomes 



d+h 



h (134). 



173. The practical rule supplied by, or derived from this equation, 

 may be expressed in words at length in the following manner. 



