In his book 'Das Saxophon', 1931, Jaap Kool describes his own observations in relation to the parabolical cone. He does not take an instrument of Sax sr. as an example but one of Sax jr. The book shows a picture of the author looking down from the neck side into the body of a saxophone. He reports to see that the tone hole side of the body has a long and gentle curve outwards just as the the opposite side, but at the same time the side walls would be perfectly straight. From top to bottom the bore would thus be progressively elliptical. As follows:
The cone that Kool observes is in fact a "cone rentrant", a contracted cone. The contracted cone actually is the opposite of the "cone parabolique", as stated literally by Sax in the 1866 patent. What Kool does is to present its opposite (!) as if it were a parabolical cone. Moreover, it is true that many saxophones prove to be out of round, but there is no well defined tendency to be found here, such as an ever increasing ovalness [in one direction] down the main tube, as would be the case if Jaap Kool's hypothesis would be correct. Besides, in most instruments any out-of-roundness has largely disappeared when we reach the joint between body and bottom bow (the joint itself adds sturdiness to the instrument).
It is very well possible that Kool justly observes that the tone holes become visible one by one and that this side of his instrument is indeed slightly curved. Many a saxophone is not quite straight because of the pulling of the tone holes and I suspect that Kool was misguided by this effect. But that the adjacent sides would be straight and that the curvature runs as Kool suggests is denied by the bore profiles. His eye is mislead by the smooth and shiny wall. Yet he thinks that these long curved walls act as a parabolical mirror that would reflect the sound outward:
Hierdurch entsteht jener eigenartig sonore, weiche und zugleich etwas hohle Klang des Saxophons, dem sich bei schlechten Instrumenten leicht eine Bauchrednerische Note zugesellt.
He makes it even more beautiful were he thinks to detect parabolic mirrors in a few spots along the tube that would reflect the sound to the outside. Especially for this purpose in these spots there would be a slightly larger tone hole. Here Kool mixes up the behaviour of a travelling and a standing wave. A standing wave is reflected onto itself at the open end of the tube and not on (arched) surfaces along the sidewalls of the tube. That cheap saxophones would sound bad, because mentioned arched surfaces would not be there, is therefore a pointless argument. Also, soundwaves would require much larger surfaces anyway to succesfully act as a mirror. (Waves require a mirror at least the size of half their wavelenght to be reflected.) Kool makes an analogy with reflections of sound in architecture. Yes, we do need the size of achitectural walls to effectively reflect sound waves, that's true.
Yet Kool's line of thought has been influential. Other writers have followed and adapted his arguments on the subject to their own use. Each time these proposals involve that the walls (at least in the straight part of) of the saxophone are gently curved lengthwise toward the outside. This inevitably produces a contracted cone. Already on a very day-to-day level on the basis of the saxophone bore profiles we can conclude that this is not the case: certainly since the days of Jaap Kool, this part is always a straight cone. Sometimes these proposals also imply that the main tube of the saxophone itself is somewhat curved, like in the sketch below, which was drawn according to John Edward Kelly's writings.
Kelly presumes that in a saxophone with a parabolical cone a benificial tension would intentionally be created between the curved wall with the tone holes and the opposite straight wall. Notwithstanding Kelly's merits as a musician, we regret to say that sound is not interested in such an unstraightness. Only short and wide bows (of the type we find in the bottom bow) have an appreciable acoustical effect. Moreover, Sax wrote in his 1866 patent
Quant à la forme, le saxophone peut en affecter diverses; mais la plus droite, celle dont le tube présente le moins des courbes, est toujours préférable.
indicating that there is no need to even search in this direction to unravel the mysteries of the parabolical cone.