WITHDRAWN FROM THE ACTION OF GRAVITY. 359 
the curve is parallel to the axis of symmetry. From s to » the radius of curva- 
ture and the perpendicular have evidently contrary directions, and the quantity 
Ae 4. ; : 
MN constitutes a difference; therefore, from one point to another of this are, 
the quantities M and N must vary in the same direction; and as the perpen- 
dicular continues to increase from the point s to the point x, the radius of curva- 
ture must continue likewise to increase; whence it follows that from s to x the 
curvature is continually decreasing. Still further on, that is from 2 to h, we 
see that the radius of curvature and the perpendicular are directed towards the 
: } 
same side, so that the two terms of the quantity MON are of the same size, and 
hence from one point to another the quantities M and N must vary in opposite 
directions. Now, when we remove from » on the are nh, the perpendicular 
begins to diminish, since at the point x itis infinite; while the radius of curvature 
begins to increase, or, in other terms, the curvature at the beginning diminishes, 
and, whatever its ulterior course, will always be, at every point of the are nh, 
weaker than at x, for at all those points the perpendicular is finite, and conse- 
quently less than at x. But we know that the curvature continues to increase 
from xz to s; therefore, in the whole extent of the are zh, the curvature is less 
than at any point of the are ns. 
Fig.29 
=< S 
Fig.30 
This being premised, let us draw the right line Ar parallel to the axis of 
symmetry, and then construct, beginning at the point , an are z¢ exactly sym- 
metrical as regards the are mv. In the whole length of the are z/ the curvature, 
by reason of what has been said, will be less than at any of the points of the 
are nt; whence it follows that this last are will be entirely interior to the former. 
Now, the are zt meets at ¢ the right line fr by an element which necessarily 
makes with the part tv of that line an acute angle; then, in order that the are 
nh, which proceeds from z in the same direction with the are wé, should meet 
perpendicularly at / the right line 47, it would be necessary that, after separating 
from the are né, it should afterwards again approach it, which is evidently im- 
possible in consequence of the inferiority of the curvature at all its points. We 
perceive, indeed, that it ought to cut the right line 27 under a more acute angle than 
does the arezf. Thus, the curve, in declining at its departure from x towards the 
axis of symmetry, cannot cease to withdraw from the axis of revolution; and since, 
moreover, it cannot change the direction of its curvature, it must necessarily 
intersect the axis of symmetry. We further perceive that, in consequence of 
the condition which governs its curvatures, it must cut that axis obliquely, so 
that we arrive, in the end, at the conclusion that it forms a node, (Fig. 30°) _ 
We shall verify the existence of this node by means of experiment. If we 
have not commenced by doing so, it is because it was necessary first to demon- 
strate that, starting from a constriction, for which the pressure is less than for a 
plane surface, there is no other form possible for the meridian line. 
