108 J. W, Davis — Motion of Compressible Fluids. 



sections of the curves represented by (2), (4), when a, XI, are 

 constants, and p, q, are variables. (2) becomes the common 

 hyperbola, which, as a increases or diminishes, approaches or 

 recedes from the intersection of the axes of p and q. One 

 branch of this hyperbola occupies the quadrant of positive or 

 physical values. To this quadrant we shall confine our atten- 

 tion. The slopes of the curves (4), (2), with respect to the 



axis of p as a horizontal axis, that of (4) being determined 

 from the second and third members of the equation, are 



dq 



1 

 QP 



dp 



dp 



dq 

 dp 



(6) 



Each curve continuously slopes downward from the axis of q 

 to the axis of p. The line 



)d P \ 



(*) 



according to (3), is either parallel to the axis of p or continu- 

 ously slopes upward ; hence it necessarily intersects (2). When 

 (7) is parallel to the axis of p, dp is the product of dp and a 

 positive constant, the first member of (1) contains the logarithm 

 of p as a factor, q in (1) or (4) is infinite where p is zero, and 

 (4) reaches the axis of q at an infinite distance ; hence (7) 



