J. W. Davis — Motion of Compressible Fluids. 109 



intersects (4). When (7) has an intercept on the axis of q, 

 one term of the first member of (1) is the logarithm of p mul- 

 tiplied by a positive constant, and (4) reaches the axis of q at 

 an infinite distance ; hence (7) intersects (4). When (7) passes 

 through the origin, it necessarily intersects (4). Therefore, (7) 

 always intersects (4) and (2), each in one point only. 



If (4) and (2) intersect, the downward slope of the former is 

 to the downward slope of the latter at the point of intersection 

 according to the following ratio derived from (6) : 



[~^](4) : [~'f](2, :: ^p'- q '- (8) 



At an intersection that occurs below the line (7), the down- 

 ward slope of (4) is greater than the downward slope of (2) ; 

 hence there cannot be more than one intersection below the 

 line (7). Likewise, above the line (7) cannot be more than one 

 intersection, and there the downward slope of (4) is less than 

 the downward slope of (2). Consequently, (4) and (2) cannot 

 intersect at more than two points. 



By variation of <r, (2) can be caused to intersect (7) at every 

 point of the latter line, and consequently at the point where 

 (4) crosses it. (4) and (2) are here tangent to each other, as 

 indicated in (7) and (8) ; and the second derivatives reduced 

 bj (7), 



d\_ _ J_ &p f ± _ 2q_ 



dp 2 ~ qp ' dp' ' dp* P 2 ' K } 



show that the contact is of the first order, and that in the 

 immediate neighborhood of the tangent point the line (4) is 

 between (2) and the coordinate axes. There can now be no 

 points of intersection of (4) and (2), since at least one of the 

 number could not conform to the law governing the slopes of 

 the curves at the intersection, and there can be no other 

 tangency, because a tangency cannot occur away from the line 

 (7). So every point of (4), except the point of tangency, is 

 between (2) and the coordinate axes. As a is increased with- 

 out limit, the tangency becomes two intersections, and these 

 continue to exist until the whole length of (4) is traversed to 

 the coordinate axes, because neither of the intersections can 

 disappear without changing into a tangency, and no tangency 

 can occur away from the line (7). The curves (4) and (2) have, 

 therefore, two intersections, or one taugency, or two imaginary 

 intersections ; and q in (5), if it has one positive value, has 

 also one other positive value, and only one other. 



Let q\ q 1 ' ', denote the alternative velocities at any point of 

 the tube. (2) and (3) show that there are two corresponding 

 alternative densities, p\ p'\ and two corresponding alternative 



